Mink$_4\times S^2$ Solutions of 10 and 11 Dimensional Supergravity

We complete the classification of Mink$_4$ solutions preserving $\mathcal{N}=2$ supersymmetry and SU(2) R-symmetry parameterised by a round $S^2$ factor. We consider eleven-dimensional supergravity and relax the assumptions of earlier works in type II theories. We show that, using chains of dualities, all solutions of this type can be generated from one of two master classes: an SU(2)-structure in M-theory and a conformal Calabi--Yau in type IIB. Along the way we uncover evidence for a new solution generating technique generalizing T-s-T, to scenarios with only a single U(1) isometry. Finally, using our results, we recover AdS$_5\times S^2$ solutions in M-theory and construct a compact Minkowski solution with Atiyah--Hitchin singularity.


Introduction
Some of the most physically relevant supersymmetric solutions of supergravity in diverse dimensions are those exhibiting a warped Minkowski factor. From early on, the main reason for this is that Minkowski vacua of string and M-theory are required to make contact with known particle physics phenomena in four dimensions, where one should arrange for the co-dimensions to be compact. Another reason, which clearly gained considerable traction with the advent of the AdS/CFT correspondence, is that all AdS solutions admit a description in terms of a foliation of Minkowski over a non compact interval -namely the Poincaré patch.
The most simple way to realize a Minkowski vacua from ten or eleven dimensions is to assume that the compact internal space accommodates some holonomy group, specifically SU(3) or G 2 for compactifications of string theory or M-theory down to four dimensions respectively. Such solutions preserve (at least) N = 2 supersymmetry and have been well studied in the literature [1][2][3][4][5], with a resurgence of the study of G 2 holonomy manifolds happening in resent years [6][7][8][9] (see also [10][11][12][13] for G 2 arising in a heterotic context). However, this kind of manifold support neither fluxes nor a warping of the Minkowski directions, so it is reasonable to expect that they represent a rather small region of the space of possible solutions. The inclusion of fluxes requires one to generalize the notion of group holonomy to group structure (G-structure) [14][15][16][17]. It is well known that no compact regular solutions of this type exist [18][19][20] -indeed a necessary element of such constructions are localized singularities, namely O-planes and their generalizations through string dualities, or lifts to M-theory (e.g. Atiyah-Hitchin singularities). Most progress constructing compact solutions with fluxes has happened by taking another restrictive ansatz, namely assuming the internal space is conformally a holonomy manifold [21][22][23], allowing one to broadly use the same mathematical tools as before. At this point all Mink d solutions for d > 1 (with the exception of Mink 2 in eleven dimensions) have been classified (see [24][25][26][27][28] for eleven dimensions, [29][30][31][32][33][34][35][36] for ten dimensions), however finding solutions beyond the ansatz of warped holonomy has proved challenging -see [38,39] for success in this direction. The issue appears to be one of tractability, so it would be helpful to have some additional guiding principle.
AdS solutions necessitate the inclusion of fluxes, early examples were also constructed by studying warped holonomy manifolds (albeit now non compact) using the Freund-Rubin ansatz [40]. By now the state of affairs for AdS solutions beyond this class is rather more developed than the Minkowski vacua cases. Many AdS solutions preserving 16 supercharges or more, when they exist, are either completely known (see the cases of AdS 7 [41], AdS 6 [42] in IIA/M-theory AdS 5 × S 2 [43] in IIB 1 ), or known locally up to solving comparatively simple partial differential equations (see the cases of AdS 6 [44,48] and AdS 4 [48] in IIB and AdS 5 × S 2 in M-theory/IIA 2 [46,49]). The exceptions are AdS 2/3 (see [51][52][53] for certain ansatz 3 ), and AdS 5 with the requisite R-symmetry not realized by a round S 2 -but these will not be our focus here. Many cases with less supersymmetry are also well studied, in particular, with the recent addition of [53], all AdS d solutions with d > 2 and at least minimal supersymmetry have been classified (see [24,28,30,[53][54][55][56][57][58]63], and the AdS 6,7 classifications above 4 ) -broadly speaking these classes have been more successfully used to find solutions beyond restrictive ansatz than their Minkowski counter-parts. The AdS/CFT obviously motivated these classifications, but in addition to the extra impetuous this provided, these classifications benefit from additional symmetry with respect to the Minkowski cases. One issue however, is most classifications for AdS, while quite detailed, assume global AdS factors from the start, so are not particularly useful for studying certain non conformal behaviors such as RG flows. For this reason it would be useful to embed the AdS classes into a more general set up 5 .
Here and in the earlier works of [49], [50], the philosophy is to learn from the successes of the AdS classifications and perform a Mink 4 classification that assumes some additional symmetry so as to enable a more detailed description than previous efforts. A good starting point for this is to classify N = 2 that preserve an SU(2) R-symmetry in the form of a round S 2 factor (see also [36] where Mink 3 × S 3 are classified) -one can then try to break some of this (super)symmetry and generate many more solutions in the spirit of [64]. An SU(2) R-symmetry is a necessary part of the super-conformal algebra in d = 5, 6 and d = 4 with N = 2. As such a corollary to this classification endeavor is that it provides an embedding of the known half-BPS AdS d solutions (with d > 4) into a broader context still preserving SU(2) R . In [49], [50] such classifications where performed in types IIA and IIB respectively, under a certain simplifying assumptions. The purpose of this work is to complete this program by classifying solutions in eleven-dimensional supergravity and relaxing the assumptions made in the earlier works.
The lay out of the paper is as follows. In section 2 we classify N = 2 Mink 4 solutions in M-theory that realize an SU(2) R-symmetry in terms of a round S 2 factor in there internal space, which leaves a five-manifold to be determined by geometric supersymmetry constraints. We find that there are two classes of solution, we refer to as A and B. Class A is governed by an SU(2)-structure in five dimensions while Class B is the M5-branes with SO(3) rotational invariance in it's co-dimensionsthe physical fields of this class also solve the supersymmetry condtions of case A, however the Killing spinor for case B is different. Additionally we make some simple ansätze for Case A, specifically a conformal CY2 ansatz and one where the internal space contains an additional squashed S 3 .
In section 3 we turn our attention to Mink 4 × S 2 in ten dimensions. We drop the assumption 2 see [47] for some explicit examples in IIA 3 Recently there was also [37] which makes no assumption beyond the existence of a time-like Killing vector 4 We don't speak of d > 7 because such AdS solutions break SUSY, however see [60] for examples of AdS8 5 See [61] and [62] for some recent progress in this direction using exceptional field theory and consistent truncation respectively of equal Majorana-Weyl spinor norm made in the [49], [50] making the classification of type II completely general. We find that all solutions that lie outside the existing classifications always have a U(1) × U(1) flavour symmetry and can be generated from a "parent" system in M-theory (the case B M5-brane) via chains of dualities. Section 4 elucidates the connection between the different classes of solution contained in [49], [50]. We are able to show that all such Mink 4 × S 2 solutions in type II can be generated from two master systems using chains of dualities -a conformal Calabi-Yau system in IIB and the SU(2)-structure in in M-theory (Case A). In the process of deriving some of the descendant cases we uncover evidence of a solution generating technique generalizing T-s-T to certain scenario containing a single U(1) isometry.
Finally in section 5 we close with some simple examples contained within M-theory case A. Specifically we establish how the N = 2 AdS 5 class in M-theory is embedded in the SU(2)-structure of case A and we point the way towards some compact Mink 4 solutions with fluxes and Atiyah-Hitchin singularities.

Mink 4 × S in M-theory
In this section we will classify N = 2 supersymmetric warped Mink 4 solutions in eleven dimensions realizing an SU(2) R-symmetry with a round S 2 .

The Spinor Ansatz
We are interested in supersymmetric solutions to eleven-dimensional supergravity with a warped four-dimensional Minkowski factor. As such we decompose the metric as where e 2∆ is a functions with support on M 7 and the four-form flux G is necessarily purely magnetic. As we seek N = 2 solutions respecting the warped product of R 1,3 × M 7 , our eleven-dimensional spinor will decompose as where ζ a + is a doublet of positive chirality Mink 4 spinors, χ a a doublet of seven-dimensional spinors such that ||χ a || 2 = e ∆ , χ a χ a = 0 (2 .3) and c denotes Majorana conjugation. It is argued in [27], that it is possible to be slightly more general than this, however we show in Appendix A that (2.2) is sufficient for our considerations. We will assume also that our spinors χ a are charged under an SU(2) R-symmetry, realized by a round S 2 factor in the internal geometry so that the metric and flux on M 7 further decompose as i.e. as a foliation of S 2 over M 5 , where e 2C is a function depending on the coordinates on the five-dimensional manifold. At the level of the spinors the SU(2) R-symmetry will be realized by decomposing our doublet of spinors as where ξ a andξ a are two SU(2) doublets formed from the Killing spinors on S 2 , ξ, which obeys where we used the Pauli matrices to represent the Clifford algebra on S 2 . As established in [49], the doublets take the form where σ 3 is the 2d chirality matrix and ξ c = σ 2 ξ * . Plugging into the the eleven-dimensional Killing spinor equation we get two independent reduced equations for the seven dimensional spinors χ 1 and χ 2 . However, given the doublet transformation property under the spinorial Lie derivative where K i are the Killing vectors of SU(2), we have that we can generate the solution for χ 2 from the solution of χ 1 , as such it will be sufficient to solve the supersymmetry conditions for the N = 1 subsector governed by the seven-dimensional spinors In the following section we shall establish a set of geometric conditions for the five-dimensional submanifold, the solutions to which are in one-to-one correspondence with all the supersymmetric solutions that follow from (2.9).

Supersymmetry Conditions from seven to five dimensions
The supersymmetry conditions for a warped product of R 1,3 × M 7 with spinors of the form were studied in [25,26], where necessary and sufficient geometric conditions were derived for the preservation of supersymmetry in the particular case χχ = 0. In the convention of [28] these are define an SU(3)-structure. In fact these geometric conditions combined with Bianchi identity of G 4 are necessary and sufficient conditions for a solution with N = 1 supersymmetry to exist [65].
Since we want the solution to be independent of the S 2 directions so as not to break the SU(2) isometry, we can extract conditions on M 5 from (2.11a)-(2.11d) by imposing (2.4) and that our seven-dimensional spinor will take the form where we η i and ∆ are independent of the S 2 directions. The first conditions we encounter comes from (2.3), given (2.13) from the first identity one finds which can only be consistent with e ∆ = χ † χ if The second condition of (2.3) imposes Let's see how the seven dimensional bi-linears decompose into products of two-and five-dimensional ones. To do this we use the following gamma-matrix representation We also decompose the five-dimensional spinors in a common basis in terms of a unit norm spinor η as η 1 = q 1 η, η 2 = q 2 (i cos αη + 1 2 sin αwη), q 2 1 + q 2 2 = 1, (2.18) which is the most general parametrization consistent with (2.15), (2.16). To calculate the forms in (2.12) we will make repeated use of the bilinear product identity where the plus and the minus indicate that we have to consider just form with even and odd degree respectively, while the presence of σ 3 depends on our parametrization of the gamma matrices (2.17).
The bi-linears that follow from η are given in [63] and read: where v, w 1 = Rew, w 2 = Imw u 1 = Reu, u 2 = Imu (2.21) defines a vielbein in five dimensions. And finally the bi-linears that following from ξ [49], where y i are coordinates embedding S 2 into R 3 and k i are one forms dual to the Killing vectors of SU(2) which may be parameterised as The first thing we calculate is the one form K = −q 1 q 2 e C cos αdy 3 + q 1 q 2 sin αw 1 + q 2 2 y 3 cos α(cos αv − sin αw 2 ) The appearance of k 3 here is opportune because dk i = 2y i Vol(S 2 ) which means that the only way to make K consistent with (2.11a) is to set the coefficient of k 3 to zero -thus we can without loss of generality take and then rotate the 5 dimensional frame such that (2.24) simply becomes K = −e C cos αdy 3 + sin αw 1 + y 3 cos αv, (2.26) In this frame the other forms become where in simplifying the last of these we make use of the identity We now plug (2.27) into (2.11a)-(2.11d) and factor out all dependence on S 2 . The result of this operation is the following set of five-dimensional form constraints Clearly the behavior is quite different depending on whether or not α = 0, so one should look at these cases separately, which we now proceed to do in the next section. Note that solving these conditions and the Bianchi identity for the flux, implies the rest of the equations of motion [34]. Before we move on let briefly examine the form of the SU(3)-structure in the internal six dimensions orthogonal to K. Any SU(3)-structure can be expressed in canonical form in terms of a complex vielbein E i as With a little effort, one can reverse engineer a complex vielbein by manipulating (2.27), we find where it is easy to confirm that this does indeed lead to a factorised metric M 7 = S 2 × M 5 We shall now proceed to classify the solutions that follow from the five-dimensional supersymmetry conditions, (2.29a)-(2.29e) -there are two cases contained in sections 2.3 and 2.4.

Case
In this case η 1 and η 2 are proportional to one another, specifically η 1 = iη 2 , and so they define a SU(2)-structure on M 5 , This means that we cannot find local expression for the vielbein u, w without any further assumptions. The purely geometric supersymmetry conditions are simply We have a solution whenever we solve (2.33), (2.33), (2.35) and the Bianchi identity of G is satisfied, which requires d(e 2C F 2 ) = 0, (2.36) away from localized sources 6 . We can solve (2.34) without loss of generality by introducing a local coordinate ρ such that Solutions in this class then take the form where M 4 supports the SU(2)-structure.
In the next subsections we shall make some assumptions about the form of the SU(2)-structure to obtain more detailed classes.

Simple Ansatz: Warped SU(2)-holonomy
The easiest way to solve this system is define the local coordinate And to impose that the part of the metric orthogonal to v has warped SU(2)-holonomy, i.e. e ∆ ω 2 =ω 2 = e ∆ j 2 =j 2 , (2.40) withω 2 ,j 2 closed. The local form of the solution is then with M 4 any SU(2)-manifold. These solutions are all non compact and the warp factor is not indicative of a simple brane set up. As suggested from [74], this can be a KK6-M2 system or something more exotic like the lift of a O6-D2 system.

Squashed S 3 Ansatz
Let's consider the simple case where the SU(2)-structure is defined on a squashed S 3 trivially fibred over an interval y. Here we take the vielbein to be where C 1 , C 2 , K, ∆ are warping function which depend on ρ = e 2∆+C and y only, h is an arbitrary function of y and ω i are the right-invariant one-form defined on S 3 , which satisfies Notice that this means we are enhancing the R-symmetry to SU(2)×U(1) as (ω 1 +iω 2 ) will generically come with a phase e iψ , where ∂ ψ is the U(1) of the Hopf fibration. From (2.34) we get the following three equations: These are solved by defining C 1 and C 2 in terms of two functions f (y) and g(y) such that At this point, because it simplifies later expressions, we choose the arbitrary function h and use diffeomorphism invariance in y to fix f as without loss of generality. This fixes the metric as where η is a potential for the Kahler form onS 2 . The two condition involving the flux in (2.35) impose that which allows one to define ∆ in terms of K. What remains of (2.35) just defines the flux F 2 , which can be used to express the M-theory 4-form as All we need to do now is solve the Bianchi identity of the flux, which away from localized sources leads to just one partial differential equation (PDE) in e K , namely This bares a striking similarity to the PDE governing the D8-D6-NS5 intersecting brane system of [73] -the only difference is the y factor. We also find deformations of this PDE in section C.2.2 of the appendix. Of course here this is only a formal similarity and the class of solutions currently under investigation is unlikely to have any relation to a Mink 6 class in massive IIA. At any rate, the form of the metric in (2.47) and that a relatively simple PDE governs the system, makes this class appear promising for finding compact solutions in M-theory 7 2.4 Case B: α = 0, The M5-brane In this case α is non zero, however one can see that it actually factors out from all of the supersymmetry conditions and in the end nothing physical depends on its specific value, as long as sin α = 0 (in which case we fall into the class of the previous section). The 1-form supersymmetry constraints are which we can be solved without loss of generality by introducing local coordinates which span an identity-structure in 5 dimensions -so we have completely local expressions for solutions in this class. The rest of the supersymmetry conditions, that are not implied by the 1-forms, involve the flux, namely The definition of F 2 can be read from (2.53) where we need to use (2.52) to define the Hodge dual: We then plug this back into (2.54) which imposes are necessarily isometries of the solution -this automatically solves (2.55) without imposing any further restriction on α, then nothing physical depends on it. Thus the local form of all solutions in this class is where the Bianchi identity of G imposes that the warp factor obeys away from localized sources. This is a five-dimensional Laplace equation expressed in coordinates that make the the SO(3) symmetry of it's solutions manifest -thus this entire class of solutions is then nothing more than M5-branes with some rotational symmetry in their co-dimensions. It is easy to check that the supersymmetry conditions (2.52)-(2.55) are actually compatible with the supersymmetry conditions (2.33)-(2.35) of case A, which means that the physical solution of Case B can actually be embedded in Case A also by taking (2.52) to define the SU(2)-structure as in (2.32).
We shall see in the next section that all solutions in type II which have internal Killing spinors with non equal norm, descend via dimensional reduction and T-duality, from this class.

Classes of Solution in type II with non equal Spinor Norm
Supersymmetric type II solutions with Mink 4 × S 2 factors were partially classified in [49,50]. Similarly to the M-theory classification of section 2, the starting point is the decomposition of the ten-dimensional Majorana-Weyl Killing spinors of the form where we have the sign + in IIB and − in IIA. Once more ξ is a Killing spinor on S 2 obeying (2.6) and now η i defines a non chiral spinor in four dimensions. The metric decomposes as where e 2A and e 2C depend on the coordinates on M 4 only. However, unlike the M-theory case, the classifications in ten dimensions [49,50] are not completely general, as we will now explain. A consequence of supersymmetry is that which means we can define constants c ± such that

4)
c + can be tuned to any positive constant without loss of generality, but tuning c − can have marked physical effects -namely the physics of solutions with generic c − is quite different to the physics when c − = 0, i.e. the case of equal spinor norm. In [49,50] it is only the equal norm case that is classified, so in this section we shall classify the solutions in type II for generic c − , or non equal spinor norm. As we shall see, it will turn out that such solutions always exhibit two uncharged isometies, so all type IIB solutions are contained in the classification of type IIA up to a T-duality, and further, all type IIA solutions have the Romans mass set to zero -so descend from our M-theory classification.
In the next section we shall study the unequal norm cases in type IIA.

Non-equal norm in IIA
We begin our analysis in IIA, our goal here is not to give a detailed classification, rather we seek to show that all solutions in this class descend from M-theory. This statement turns out to actually be true for any warped Mink 4 solution -so we will not actually assume an S 2 factor here, merely a decomposition of the metric of the form and likewise for the fluxes, with all physical feilds supported by M 6 only.
In the conventions of [66], supersymmetry depends on the existence of a non-chiral IIA spinor on warped Mink 4 × M 6 that takes the form where ζ + is a positive chirality spinor in four dimensions, χ a non-chiral six-dimensional spinor in and where the c superscript labels Majorana conjugation 8 such that c = . Supersymmetry is then implied by the following spinorial conditions on M 6 where Φ is the dilaton,γ (6) the chiral operator and where in (3.7c) we correct a typo appearing in [66]. Here F n labels the RR flux magnetically coupled to a space-filling brane One can without loss of generality take B (10) = I ⊗ B (6) where B (6) B (6) * = I as the relevant intertwiner so that however, as we shall see shortly, the 0, 2, 4 forms will turn out to be purely magnetic so we need not worry about this distinction for long. Using (3.7a) and (3.7c) it is possible to establish that d(e −A |χ| 2 ) = d(e A χ †γ χ) = 0, while including also (3.7b) one can establish that d(e 2A−Φ χχ) = 0, which one can use to fix where the final condition follows in general whenever (γ (6) ) T = −γ (6) . c ± , c are constants, we can set c + = 2 without loss of generality, c − = 0 is the cases of equal internal spinor norms and c parameterizes the G-structure of the internal 6 manifold; in particular, c = 0 corresponds to an orthogonal SU(2)-structure. Using the relations which actually hold for Hermitian gamma matrices in any dimension, we get that the rank 2,3,6 forms which come from the bi-linear χ ⊗ χ † and the rank 1,2,5,6 forms constructed fromγ (6) χ ⊗ χ † are purely imaginary, while the rest are real. Using this fact, it is possible to show from the innerproducts of (3.7a)-(3.7b) with χ andγ (6) χ that which come from the imaginary and real parts of the above inner products respectively. Equal spinor norm is equivalent to χ †γ(6) χ = 0 in this language, so it is clear that Romans mass do not vanish only for equal spinor norm. As such all type IIA solutions with non equal spinor norm can be lifted to M-theory As we shall show later in section 3.3, the Mink 4 × S 2 non equal norm solutions in IIA necessarily descend from section 2.4 by dimensional reduction. In the next section we turn our attention to type IIB solutions with non equal norm, as we shall see, this requires a more detailed analysis.

Non equal norm in IIB
In this section we study the Killing spinor conditions in IIB. Our goal is to establish that when the spinor norms are not equal, there are always two flavour U(1) isometries on the solutions -however, most of the results in this section apply irrespective of the particular values of the norms.
In IIB we shall define our ten-dimensional Killing spinors as in 3.1, but redefine the 4d spinors asη As we establish in Appendix B, supersymmetry imposes the following set of four-dimensional spinoral constraints Where f, g, H 1 , H 3 are the form fields defined as in Appendix B while Φ, C, A are function on M 4 . From the six-dimensional zero-form conditions (B.5), imposing that the physical quantities are SU(2) singlets leads to the following equations for the four-dimensional spinors: Working a bit harder we can establish another scalar condtion from the difference between η 2 (2(3.13a)+2(3.13e)-(3.13c)) and η 1 (2(3.13b)+2(3.13f)-(3.13d)) 9 , namely One can solve all of these scalar conditions with the following decomposition of η i in terms of a single spinor η, and complex functions a, b so that α parametrise the difference in norms of the spinor, which are equal only when cos α = 0. 9 We define η = (η c ) † where η c = Bη * and the conventions for B are given in Appendix B.
As shown in [36,44], the spinor η is sufficient to define an identity-structure on M 4 , which is expressed in terms of the vielbein The definition of w is such that η c = 1 2 wγη , which we can use to fix the phase of b in (3.16) by rotating the vector w,so we choose b to be real and expand a as Let us now consider the vector which can be rewritten as a one form in terms of the vielbein (3.18) We will spend the rest of this section to proving that k is a Killing vector parameterizing a U(1)×U (1) flavour symmetry in the solutions. However we shall first perform some consistency checks: We notice that for generic values of α both real and imaginary components of k cannot vanish, so k is necessarily non zero. The exception is when cos α = 0, which is the case of equal chiral spinor norm studied in [50] -here the number of isometries we find depends on the values of a 1 , a 2 , b. In particular there is the following reduced isometry structure when the functions of the spinor ansatz are tuned as follows 0 isometries: cos α = a 1 = b = 0, 1 isometries: cos α = a 1 = 0, b = 0, which can happen because k is respectively zero or purely imaginary in these two cases only -so k is non zero cos α = 0 and is also consistent with the isometry structure of known solutions with equal norm, a promising sign we are on the correct track.
The first thing to establish is that k is an isometry with respect to scalar physical fields. To do so one can take the sum of the inner products of (3.13a), (3.13b) withγη 2 ,γη 1 respectively and repeat this operation for (3.13c),(3.13d) and (3.13e), (3.13f). This leads to the conditions k a ∂ a A = k a ∂ a Φ = k a ∂ a C = 0 (3.22) which means that A, C, Φ do not depend on k directions. Now let's prove that k is Killing with respect to the metric, i.e., that ∇ (a k b) = 0. A straightforward calculation leads to: and by the manifest anti-symmetry of the right hand side, it is immediate to notice that therefore k is a Killing vector. This expression can be further simplified, indeed, using the combinations of equations (2(3.13a)+(3.13e)-(3.13c)) and (2(3.13b)+(3.13f)-(3.13d)), which read one may establish from the sum of η 2 γ abγ (3.23) and η 1 γ abγ (3.24), that Therefore we get This expression is particularly useful because allows us to establish that the isometry group is U(1) × U(1). To prove this we need to show that the Lie bracket [k, k] = 0 which, in the case at hand, is equivalent to k a ∇ a k b ∈ R.
It is then a simple matter to calculate that which is clearly real -so the isometry group of k is indeed U(1) × U(1). The next task is to prove that U(1) × U(1) is a flavour symmetry -which means that the spinors η 1,2 are not charged under these isometries. This statement can be caste as the vanishing of the spinoral Lie derivative along k, ie To prove this we will need the following algebraic relation, which can be demonstrated using standard Fierz identity techniques: where k and v 2 must now be understood as gamma matrices (via Clifford map). Let us fix i = 1 (the discussion for i = 2 is specular) and start by evaluating the first term in (3.30). Contracting (3.13g) with k a and using the first two relation of (3.31) one finds: Now we can substitute for the RR fluxes using (3.23) multiplied byγ. With some manipulations one realizes The term k a ( H 3 + iH 1 ) a can be evaluated from the inner product of (3.13c) with η 1 and (3.13d) with η 2 , indeed this leads to note that is implies that when the norms are not equal, so that k defines two Killing vectors in general, H 1 has no legs in the isometry directions, while H 3 has a leg in each. We are left with the expression Now we consider the second term in (3.30): Using (3.26) one may write Since k a is orthogonal to ∂ a (2A+C −φ) and (v 2 ) a , we can take γ ab = γ a γ b in the previous expression and, using (3.31) we find Now one simply combines the contributions form (3.35) and (3.37) to establish that It is easy to perform the same steps for L k η 2 and therefore we get that U(1) × U(1) is a flavour symmetry. The last step is to prove that also the fluxes are not charged under the isometry. Extrapolating from the classification of [30], we know that the RR fluxes are defined in terms of bi-linears of η i and the NSNS 3-form. Since the spinors are uncharged, the same follows for the bi-linears and so we need only establish that the NSNS 3-form is a U(1) × U(1) singlet. We begin by considering η 1 (3.13a), η 1 (3.13c) + η 2 (3.13d), η 2γ (3.13a), η 2γ (3.13e); these, together with (3.34), allow us to find the following relations: (3.39) The first, with the Bianchi identity d(e 2C H 1 ) = 0 implies L k (e 2C H 1 ) = 0. Now let's consider the equation of motions for the B-field, this implies the following equation Contracting this with k and using (3.39) we get that which proves that H 1 and H 3 are singlets under U(1) × U(1).
We have now established that all non Mink 4 × S 2 solutions in type IIB with non equal spinor norm necessarily have a U(1) × U(1) flavour symmetry. In the next section we shall show that they all actually follow from IIA via T-duality and can then be lifted to the class B in M-theory.

Relation to the M5-brane
We have proved that all supersymmetric Mink 4 × S 2 solutions with non equal spinor norm have an uncharged U(1) × U(1) isometry when they are in type IIB and can be lifted to M-theory when they are in IIA. We shall now prove that they all descend from Case B in M-theory -which is the M5-brane. First let us establish that Case B implies the IIA solutions.
Whenever one has a U(1) isometry in M-theory one can reduce down to type IIA along it. When one reduces from eleven dimensions to ten, it is convenient to choose a frame such that where e a d is the vielbein in d space-time dimensions, ∂ ψ is the isometry direction, Φ the dilaton and C 1 the potential of the RR 2-from in IIA. When one reduces, the gamma-matrix corresponding to e 10 11 then becomes the chirality matrix in in IIA. The non equal norm condition in IIA, in the conventions of section 3.1, is equivalent to χ †γ(6) χ = 0 and, up to an overall warping not relevant for our discussion, the six-dimensional non-chiral spinor in IIA and seven-dimensional spinor in Mtheory are the same. The non equal norm condition in M-theory therefore reads χ † γ (7) m χ(e 10 11 ) m = K m (e 10 11 ) m = 0. Therefore we can establish whether or not we reduce to a non-equal-norm class in IIA by studying the M-theory 1-form K defined in (2.26). Specifically, we can generate non equal norm in IIA whenever K has a leg in the isometry direction we reduce on.
There are two ways to get a round S 2 factor in IIA via dimensional reduction of a parent solution in M-theory. 1) The parent solution has a round S 2 and a U(1) isometry orthogonal to this to reduce on. 2) The parent has a U(1) fibred over S 2 , ie an S 3 that may be squashed along the Hopf fibre direction only. We can immediately rules out option 2) as in this case K ∼ dψ + η + ..., with dη = Vol(S 2 ), which can never solve the 1-form condition place d(e 2∆ K) = 0 -as such IIA non equal norm must descend from Case A or B in M-theory. Here the only place 10  that will give rise to non equal norm in IIA is w 1 . As w 1 is only turned on when sin α = 0, we can realise all non equal norm solutions in type IIA through reduction from Case B in section 2.4, which is simply the M5-brane class with SO(3) rotational symmetry in it's co-dimensions, which is rather remarkable. The local form of this solution is given in (2.58), where giving w 1 a leg along the reduction isometry is equivalent to making the local coordinate ∂ x 1 and isometry without loss of generality 11 . Together with the two aprior U(1)'s in section 2.4 (packaged as part of a Mink 6 factor there) this gives three flavour U(1) isometries, which implies that two will always remain when we reduce to IIA, just as we found in IIB class.
We have shown that all non equal norm solutions in IIA descend from Case B in M-theory and all such type II solutions come with a flavour U(1) × U(1) isometry. Naively, one might then conclude that all the IIB solutions follow from IIA via T-duality -but there is another possibility to rule out. Much like option 2) for the M-theory reduction, round S 2 in IIB could follow by T-dualizing on the Hopf fibre of a squashed S 3 . Such a solution in IIB would necessarily have but fortunately for us, this was already ruled out for non equal norm in the previous section by the necessary condition k a (H 1 ) a = 0. We have shown that the parent or master class of all non equal norm Mink 4 × S 2 solutions in type II is an M5-brane with SO(2) × SO(3) rotational symmetry in it's co-dimensions. All solutions 11 This is not the only way to form a U(1) to reduce on, in general one could take and impose that ∂ ψ is an isometry -however in IIA ci can be turned off with a coordinate transformation and rescaling of gs. One can also form a U(1) of the metric and flux by expressing x1, x2, in polar coordinates, however this is not an isometry of the G-structure, as (2.26) makes clear. Indeed such an isometry is charged, so reducing on it would brake supersymmetry.
in IIA and IIB are then generated from this by a chain of dualities depicted in Fig 1. However, we should stress this does not mean that the type II classes are completely trivial -there are several distinct ways to reduce to type IIA on a U(1) subgroup of the available U(1)×U(1)×U(1) uncharged isometry, and yet further ways to T-dualize inside the residual U(1) × U(1) to get to IIB -the result of this chain of dualities can potentially end up being rather complicated. Rather we would just like to stress that if one is interested in constructing a non equal norm Mink 4 × S 2 solution in type II, this is best done from the M5-brane perspective.
In the next section we shall provide master classes for the remaining Mink 4 × S 2 solutions in type II, namely those with equal spinor norm.

Master Classes for Mink × S 2 and Hint of a New Solution Generating Technique
In [49,50], Mink 4 × S 2 solutions with equal spinor norm where classified. They fall into three distinct classes characterised by the minimum number of uncharged U(1) isometries the internal four-manifold M 4 contain a priori 12 . Specifically one has Class I has no a priori isometries on M 4 , for Class II one generically has a U(1) bundle over a 3-manifold, while generic solutions in Class III have a T 2 bundle over a 2-manifold -They can be found in sections 4.3, 4.4 and 4.5 of [49,50] for IIA and IIB respectively. Cases II and III contain 1 and 2 constant parameters respectively -when they are set to zero one reduces to classes that have local Mink 5 × S 2 and Mink 6 × S 2 factors. More surprising is that all solutions in class III, all in IIB class II, and many in class II IIB, are governed by the same PDE's irrespective of whether or not these parameters are turned on -such generic solutions in these classes can be viewed as parametric deformations of un-fibred Mink 4 × S 1 × S 2 and Mink 4 × T 2 × S 2 solutions respectively, where the first two factors share a common warping in each instance 13 . The existence of parametric deformations raise an obvious question -is some sort of duality at play? Two obvious candidates are formal U-dualities of the type in [69] and T-s-T transformations [70] -which are both solution generating techniques involving chains of string dualities and coordinate transformations (specifically shifts between U(1) isometries) that do not commute with 12 This in turn is related to which inner products of the two independent non-chiral four-dimensional spinors are assumed to be non vanishing, see [49,50] for details. 13 When this is true, locally, there is no difference between Mink4 × T n and Mink4+n. However, for the parametric deformations the warp factors are no longer the same and T n becomes fibered over a base. As such it makes more sense physically for only the Mink4 directions to be non compact in these cases.
these dualities. As we shall see this is enough to explain case III, but case II is more subtle and seems to point the way to a new solution generating technique.
In the previous section we established that all solutions in type II supergravity with non equal norm can be obtained from the M5-brane, by reduction and T-duality -so one can view the M5brane as a master class for all such solutions. In this section, in addition to explaining the origin of the parametric deformations, we establish master classes for the non equal norm in type II. The first of these is M-theory Case A in section 2.3, the second is class I in IIB -note that after imposing a U(1) isometry in the M-theory class and reducing on it one end up in class I in IIA. We shall quote these solutions in the next sub-sections for convenience. All other cases can be generated from these via certain chains of dualities, as we shall show -the maps between solutions are also summarized in Fig 2.

Master Class in Type IIB: Case I in IIB
The master class in Type IIB has a NS sector that takes the local form where e 2A depends on all four-dimensional local coordinates x i , and f, g have support on Σ 2 only. The RR sector is generically non trivial, with F 1 , F 3 , F 5 all turned on, and can be found in [50]. Solving the Bianchi identities of these fluxes (away from localized sources) requires the following PDE's to be solved The six dimensions orthogonal to Mink 4 in-fact support a conformal Calabi-Yau. Clearly if impose that one of ∂ x 3 , ∂ x 4 is an isometry (which g does not depend on ) and T -dualize on it we end up with a local Mink 5 class in IIA which exhausts this class. If we make both isometries (and g constant) then T-dualize on both we exhaust the Mink 6 × S 2 solutions in IIB. Of course one still has a U(1) isometries when g is an polynomial of order one -when this is the case we can gauge transform such that B respects the isometry, but as a leg in it -this generically leads to the U(1) becoming fibred over the S 2 in the T-dual.

IIA Reduction of Master Class in M-theory: Case I in IIA
Case I in IIA can be generated from M-theory by imposing a U(1) isometry within the SU(2)structure of the M-theory class in section 2.3 and reducing on it. This class is rather more complicated than it's IIB counter-part, with the six-dimensional space orthogonal to Mink 4 supporting an orthogonal SU(2)-structure. The local form of the NS sector in this class is where e 2A , e Φ , B 0 each generically depend on all the four-dimensional local coordinates x i . Supersymmetry additionally requires the following PDE's to be solved The RR sector is rather involved, but can be found in [49], however there is no Romans mass.
Ensuring that the fluxes obey the correct Bianchi identities imposed another set of PDE's We can generate the Mink 5 and Mink 6 cases in IIB and IIA respectively by imposing that the canonical coordinates on Σ 2 are isometry direction, and that dB has no leg in this isometry. Clearly if impose that one of ∂ x 3 , ∂ x 4 is an isometry (which g does not depend on ) and T -dualize on it we end up with a local Mink 5 class in IIA which exhausts this class. If we make both isometries (and g constant) then T-dualize on both we exhaust the Mink 6 × S 2 solutions in IIB.

Generating Case II
In this section we shall establish how case II in IIA and IIB are generated from the master systemsas we shall see this is achieved through a generalization of a T-s-T transformation. Generic solutions in this class are parametric deformations of solutions with metric factor ds 2 = e 2A ds 2 (Mink 4 ) + ds 2 (S 1 ) + e 2C ds 2 (S 2 ) + ... (4.6) and similarly for the fluxes and for many cases, the deformed solutions are governed by the same PDE's as the undeformed ones. The deformations are reminiscent of T-s-T transformations, but since we only have the flavour U(1) of S 1 to work with this cannot be what is happening here. The deformations turn out to be possible precisely because these solutions, after T-dualizing, can be embedded in more general master systems for Mink 4 × S 2 with no a prior isometry. One generates solutions of the form (4.6) from the master systems by imposing that one direction in Σ 2 , ∂ x 4 say, is an isometry and T-dualising on it. One can generate deformations by performing a diffeomorphism mixing ∂ x 4 and a direction in M 2 , ∂ x 1 say, in terms of an arbitrary constant c 0 ie Only then does one impose that ∂ x 4 is an isometry. Such a diffemorphism will require a frame rotation (with respect to the un-transformed case) to put the solution in the canonical T-duality frame, where the solution can be expressed in terms of a U(1) fibration over a nine-dimensional base, i.e.
If the RR fluxes have components orthogonal to x 1 but with a leg in x 4 , such a diffeomorphism and frame rotation will turn on additional RR flux components in vielbein frame that lie orthogonal to the isometry vielbein e x 4 = eÃ(dx 4 +A). Under T-duality a RR flux component parallel to e x 4 is mapped asF n →F n−1 while one that lies orthogonal is mapped toF n+1 , as such these new flux components cannot be turned off with a coordinate transformation in the T-dual, so this diffeomorphism does not commute with T-duality in this instance. One also has a non T-duality commuting diffeomorphism whenever the warp factors of the x 1 and x 4 directions in the metric are neither constant nor equal 14 .
A non T-duality commuting diffeomorphism then generates a parametric deformation in the T-dual of the master system. The result is much like a T-s-T transformation of the (4.6) class of solutions, but only a single U(1) is required. Before explicitly spelling out exactly how case II in IIA and IIB are realized from the master systems, let as pause to stress a few points. First, if x 1 and x 4 are both set to isometry directions, after the diffeomorphism, the result of T-dualizing on x 4 coincides with performing a T-s-T transformation on these directions in the (4.6) subclass of class II, so what we have here is a generalization of T-s-T. Second, clearly we were able to spot this generalization because we have completely local expressions for case I in IIA and IIB -we do not however expect this solution generating technique to require this in general.

Case II in IIA
It was already observed in [49] that when one T-dualizes on the U(1) isometry of Case II in IIA , one is mapped to a conformal Calabi-Yau type system -this system is clearly case I in IIB. To see this we perform the coordinate transformation in (4.1), the NS sector then takes the form 14 which is why we choose to use x1 rather than x3 as the additional coordinate, why we don't use x2 should be obvious.
where b 2 + a 2 2 = 1, if we now impose that ∂ x 4 is an isometry, we have and the PDE's governing the system become i.e. exactly what one has for case II in IIA. If we now T-dualize on ∂ x 4 the effect of (4.8) can no longer be turned off with a coordinate transformation, (4.9) is mapped to the NS sector of case II in IIA (see sec 4.4 of [49] for a comparison) and the the RR fluxes and PDEs also follow. When x 3 is also assumed to be an isometry (4.12b) is the same for all values of c 0 , this is of course because the procedure reduces to T-s-T acting on the c 0 = 0 case. This is not the only way to make this happen though, indeed if b = constant, we can remove it from the PDE by rescaling x 1 , the result is another parametric deformation, with the same PDE's for all c 0 , but with x 3 no longer an isometry.

Case II in IIB
We can generate case II in IIB from case I in IIA, which itself descends from case A in M-theory by dimensional reduction. Starting from the NS sector in (4.3) on performs the coordinate transformation and imposes that ∂ x 4 is an isometry -so that the that the physical fields depend on the local coordinates (x 1 , x 2 , x 3 ) only. Notice that the warp factors of the dx 2 1 and dx 2 4 directions are not equal, so this transformation will not commute with T-duality on ∂ x 4 generically. This time after T-dualizing , one generates case II in IIB and one can map the PDE's for genetic c 0 to those for c 0 = 0 by rescaling x 1 as in before, but without making further assumptions about the physical fields. Thus, the entire of case II in IIB is a one parameter family of solutions governed by common PDE's. To make contact with the conventions of section 4.4 of [50] one should identify the deformation parameter c 0 with the following combination of functions there where κ 2 + κ 2 ⊥ = 1 are angles parameterizing a point depended SU(2)-structure there -it is not hard to check that the right hand side of this expression is indeed a constant.
For now we are at a loss as to why the generalized T-s-T procedure does not modify the PDE's for case II in IIB, but generically does in IIA. It is of course possible that a change of coordinates exists in IIA after which the PDE's actually do match, but we were so far unable to find it. We hope to study the possible utility of the generalized T-s-T procedure as a solution generating technique in the future.

Generating Case III
In this section we will sketch how Case III in type IIA and IIB can be generated from one of the master systems. As this only involves standard solution generating techniques we will be less detailed than Case II.

Case III in IIB
Case III in IIB can be generated from case I in IIB by first imposing that Σ 2 = T 2 so that one has two isometries to work with and setting g = 0. One needs to supplement this by rescaling the dilaton, Minkowski and local coordinates, but after doing this carefully one is mapped to Case III in IIB.

Case III in IIA
Case III in IIA can be generated in a similar fashion. We again fix Σ 2 = T 2 and T-dualize on both U(1)'s therein. We are mapped to a Mink 6 system studied in [73] (see also  Along the way some additional rescaling of coordinates is required, but this gives the general ideaie Case III in IIA is generated by a combination of U-duality and T-s-T transformation.

Examples
In this section we give some simple examples that follow from case A in M-theory, we exclude case B because it is very simple -of course AdS 7 × S 4 /Z k can be embedded here, but as this just requires solving a simple Laplace equation, we omit the details. In section 5.  [46] (which was shown to be exhaustive in [71]), in this section we will show how they are embedded within M-theory class A. The metric of these solutions is of the form where D is a function of y, x 1 , x 2 and ∂ ψ is an isometry; this one together with the S 2 factor realize the SU ( In order get a metric that actually respects the isometries of AdS, we need the radial component to point along one of the vielbein that make up the SU(2) in such a way that the metric has no dr cross terms, and has a common warp factor for all the putative AdS directions -the SU(2) structure should also be charged under ∂ ψ . After some work, one is able to establish that the following four-dimensional vielbein satisfies all the SU(2)-structure conditions u = −∂ y D y e λ+ 1 2 D (dx 1 + idx 2 ), w = 2e iψ e −2λ y −∂ y D dr + 1 2 ∂ y Ddy + i(dψ + V ) , (5.5) provided that D satisfies the Toda equation away from localized sources. The flux F 2 that follows from this is then which reproduces the correct M-theory 4-form through G = e 2C Vol(S 2 ) ∧ F 2 and obeys the Bianchi identity.

Towards Compact Mink 4 in M-theory with Fluxes
The ansatz of section 2.3.2, contains a squashed S 3 in addition to the Mink 4 × S 2 factor generic to all solutions we considered. As such it is a prime candidate for solutions which support a Taub-NUT or Atiyah-Hitchin manifolds. Metrics with Atiyah-Hitchin singularities (which are the lift of O6 planes) are a possible option to make Minkowksi vacua in M-theory with non trivial fluxes compact.
Here we show that it is indeed possible to find metrics with such singularities inside section 2.3.2 with a simple example. The solutions in section 2.3.2 are defined in terms of a single undetermined function e −2K that is governed by a relatively simple PDE in two variable (ρ, y), namely (2.49). The simplest way to make progress with this PDE is with a separation of variables ansatz e −2K = p(ρ)q(y). (5.8) The PDE then reduces to two ODE's of the form where c, c 1 are constants. These ODE's are still somewhat non trivial to solve, so in principle one can could proceed numerically, however our aim is just to show the plausibility of finding compact solutions with non trivial fluxes in the squashed S 3 ansatz, so let us just make the most brutal non trivial assumption we can, namely c 1 = 0, p = 1. (5.10) As a consequence of this q = L y − b (5.11) for L, b constants and we have G = 0 while the part of the metric spanned by (ρ, S 2 ) becomes simply R 3 or equivalently T 3 . So the metric for this solution is given (for L = 1) by ie a squashed S 3 foliated over an interval -which is in fact bounded. One can check that close to y = −b the metric exhibits an Atiyah-Hitchin singularity while as y approaches −c the warp factor ofS 2 becomes constant and the remaining directions vanish regularly as R 2 in polar coordinates -thus if we assume −c < −b the interval is bounded between −c < y < −b and the manifold is compact. This example of course has no fluxes turned on, but it seems likely that, even within the separation of variables ansatz, it will be possible to find similar solutions, with G = 0 and with T 3 → (ρ, Vol(S 2 )) with the metric warped in terms of ρ. It would be interesting to pursue this -but such a detailed study is outside of the scope of this work.
where G is defined in (2.1). The first step is to perform a decomposition from 11 to 4 + 7 dimensions where ζ + is a four-dimensional spinor with positive chirality, ζ c + = ζ − and χ is a seven-dimensional one.
Evaluating (A.1) on Mink 4 directions we get the following algebraic condition: while taking the internal index where we have chosen γ 5 = iVol 4 . Multiplying (A.3) by γ a /4 we can further simplify (A.4): From these conditions we can easily prove that f = 0, indeed contracting (A.3) by χ † and make the difference with the conjugate of the same expression of we get since the bilinears of χχ † are real if they have degree 0, 1, 4, 5 and purely imaginary otherwise. When χ c = χ, we can only define a G 2 structure on M 7 . From the sum and the difference of (A.3) with its conjugate we have that F χ = 0 and ∂ a ∆γ a χ = 0. From the second condition we have: and then we can set F = 0. So the internal 7-manifold actually has G 2 holonomy and no warping factor on Mink 4 . Although such solutions may exist with M 7 =S 2 ×M 5 we need not consider them explicitly; indeed there are no Killing spinors on S 2 such that ξ = ξ c , and there is no invariant form on S 2 that maps the Killing spinor to it's Majorana conjugate -as such imposing χ c = χ makes χ contain two separable five-dimensional systems, one coupling to ξ one to ξ c -this just imposes additional constraints on the five-dimensional system that follows from χ c = χ, so cases with G 2 holonomy are special cases of the systems we consider in the main text. We thus restrict ourselves to SU(3) structure case χ = χ c . Now let's analyze the zero-form constraints on M 7 : and so we can set, without loss of generality, ||χ|| 2 = e ∆ . Moreover, we can also calculate This equation tells us that the phase of χχ is constant and then we can set it to zero choosing the following parametrization: where c is a real constant. Let's analyze the two zero-form equations using our ansatz (2.9). The first condition in terms of the five and two dimensional spinors reads: to preserve the SU(2) R-symmetry the warp factor cannot depend on the coordinate on the sphere, so me have On the other hand (A.12) reads − (y 1 − iy 2 )η 1 η 2 = ce −2∆ (A. 16) and then we must have These considerably simplify the supersymmetry conditions, and in particular we can restrict to (2.11), as we have done in the main text.

C Towards Solutions for the General IIA case
In this appendix we will find a large ansatz that allows us to simplify Case I in IIA in such a way to have a diagonal metric. In this case supersymmetry is implied by the following equations: It turns out to be useful define the following functions: The other equations of (C.1) imply that f = f (x 1 , x 2 ), g = g(x 1 , x 2 ) and µ = µ(x 1 , x 3 , x 4 ). Moreover we have the following PDEs: The fluxes are B 2 =gVol(S 2 ), (C.6a) and the Bianchi identities for these fluxes impose the following PDEs: We can notice that (C.7a) is just a restriction on the coordinate dependence of the physical fields, while (C.5) can be used to define H. If this is the case, then we can define f from this equation 2 ∂ x 2 f ) + ∂ x 1 (e −µ ∂ x 1 (f e −4A−µ )) = 0 (C.8) which is redundant whenever g gets defined rather than it's derivatives. At this point, solving these PDEs in full generality is hard; however, this class of solutions is large enough to allow us to impose some further ansatz and still obtaining interesting intersecting-brane systems.
The first thing that needs to be addressed is that the LHS of (C.5b) is independent of (x 3 , x 4 ) which means the RHS should also be, which is not true a priori. We can deal with this by making an ansatz for e −4A , we find that achieve the stated aims 16 . We will look at these cases separately in the following subsections.

C.1 Case I
Here (C.5a)-(C.5b) implies The first thing we need to do is solve (C.7a) which becomes The obvious way to make progress with this is to do one of the following f 0 = 0, f 1 = 0, or f = c 0 + c 1 x 2 for c i constant. (C.12) The first two of these turn out to be sub cases of the last up to a redefinitions of x 1 so we without loss of generality set , e µ =S(x 1 , x 3 , x 4 ), f (x 2 ) = c 0 + c 1 x 2 (C. 13) and then (C.7a) PDE's becomes a Youm like condition ∂ x 1S ∂ x 2Ũ = 0. (C.14) 16 In principle one can add a generic function e µ 0 (x 1 ) in front of the expressions for e −4A and e µ in the second case, but it is easy to check that this can be reabsorbed into f and performing a change of coordinates. Another non-trivial generalization of the second case is to define e −4A = f −1 [H(x1, x2)S(x3, x4) 2 + T (x2, x3, x4)]. This case however seems to require further assumptions which so far didn't lead to something interesting. and then the remaining PDE's become 2 U = 0, 2 S + U ∂ 2 x 1 S = 0, (C.20) The physical fields are defined as: We thus have a system of D4-D6-NS5 branes where the NS5 is smeared along x 1 while the D4 along x 2 .

C.2 Case II
Here (C.5b) becomes ∂ x 2 g = x 2 2 e −µ 0 ∂ x 1 H (C. 22) The discussion turns out to be different if we consider the warping A to depend or not from x 1 .