Fivebranes and resolved deformed G_2 manifolds
aa r X i v : . [ h e p - t h ] A p r September 18, 2018
Fivebranes and resolved deformed G manifolds J´erˆome Gaillard and Dario Martelli Swansea University,Singleton Park, Swansea SA2 8PP, United Kingdom Department of Mathematics, King’s College, LondonThe Strand, London WC2R 2LS, United Kingdom
Abstract
We study supergravity solutions corresponding to fivebranes wrapped on a three-sphere inside a G holonomy manifold. By changing a parameter the solutionsinterpolate between a G manifold X i ∼ = S × R with flux on a three-sphereand a distinct G manifold X j ∼ = S × R with branes on another three-sphere.Therefore, these realise a G geometric transition purely in the supergravitycontext. We can add D2 brane charge by applying a simple transformationto the initial solution and we obtain one-parameter deformations of warped G holonomy backgrounds. These solutions suggest a connection between the N = 1Chern-Simons theory on the fivebranes and the field theory dual to D2 branesand fractional NS5 branes, transverse to the G manifold S × R . ontents G holonomy manifold S × R G holonomy metrics . . . . . . . . . . . . . . . . . . . . . 5 S × R G solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Ansatz and BPS equations . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 One-parameter families of solutions . . . . . . . . . . . . . . . . . . . . 143.3.1 Expansions in the IR . . . . . . . . . . . . . . . . . . . . . . . . 153.3.2 Expansions in the UV . . . . . . . . . . . . . . . . . . . . . . . 163.3.3 Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 G structures 25 G holonomy solutions . . . . . . . . . . . 29 SU (2) invariant one-forms 34B Derivation of the BPS equations 36C Supersymmetry conditions in Type IIA 39 C.1 Reduction from d = 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 39C.2 Killing spinor ansatze . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 Introduction
Calabi-Yau manifolds have played an important role in string theory for many years.For example, string theory compactified on a Calabi-Yau manifold gives rise to aneffective four-dimensional theory with unbroken supersymmetry. In the context ofthe AdS/CFT correspondence [1], conical Calabi-Yau three-fold singularities equippedwith a Ricci flat metric give rise to supersymmetric solutions of Type IIB string theorywith an AdS factor [2, 3, 4, 5], thus providing precise gravity duals to a large class ofsupersymmetric field theories.The simplest and most studied Calabi-Yau singularity in string theory is the conifold[6]. There are two distinct desingularisations of this, in which the singularity at thetip is replaced by a three-sphere or a two-sphere. These are referred to as deformation and resolution , respectively. The relevance of the transition between the resolved anddeformed conifold in string theory was emphasised in [7]. The deformed conifold geom-etry underlies the Klebanov-Strassler solution [8], which is dual to a four-dimensional N = 1 field theory displaying confinement. A different supergravity solution dual to aclosely related field theory was discussed by Maldacena-Nu˜nez [9, 10]. This arises asthe decoupling limit of a configuration of fivebranes wrapped on the two-sphere of theresolved conifold.A solution of Type IIB supergravity that contains as special cases the Klebanov-Strassler and the Maldacena-Nu˜nez solutions was constructed in [11]. This was inter-preted as the gravity dual to the baryonic branch of the Klebanov-Strassler theory,with a non-trivial parameter related to the VEV of the baryonic operators [12, 13]. Itwas later pointed out in [14] that the solution of [11] is related to a simpler solution [15]corresponding to fivebranes wrapped on the two-sphere of the resolved conifold, with-out taking any near-brane limit. In this context, the non-trivial parameter can roughlybe viewed as the size of the two-sphere wrapped by the branes. When this is very large,the solution looks like the resolved conifold with branes on the two-sphere and whenthis is very small, it looks like the deformed conifold with flux on the three-sphere.Therefore, it displays an explicit realisation of the geometric transition described in[16]. The key fact that allows the connection of the resolved and deformed conifoldat the classical level is that the solution is an example of non-K¨ahler, or torsional ,geometry [17, 18]. For related work see [19].In this paper we will present a G version of the picture advocated in [14]. Inparticular, we will discuss supergravity solutions which correspond to M fivebranes2rapped on the three-sphere inside the G holonomy manifold S × R [20, 21], withouttaking any near-brane limit. Solutions of this type were previously discussed in [22].If we take the near-brane limit, we find the solutions discussed by Maldacena-Nastasein [23], based on [24]. These were argued in [23] to be the gravity dual of N = 1supersymmetric U ( M ) Chern-Simons theories in three dimensions, with Chern-Simonslevel | k | = M/ G holonomy metrics on S × R comprises three branches, that we will denote X i [25], intersecting on the singularcone. These three branches are related to the three branches of the conifold modulispace, namely the deformation, the resolution and the flopped resolution [26, 25]. Onetherefore expects close analogies with the discussion in [14]. Indeed, by working in thecontext of torsional G manifolds [27], we will find a set of one-parameter families ofsolutions that pairwise interpolate between the three classical branches of G holonomy.The non-trivial parameter in these solutions can roughly be viewed as the size of thethree-sphere wrapped by the fivebranes. When this is very large, the solution lookslike a G manifold X i with branes on a three-sphere and when this is very small, itlooks like a distinct G manifold X j ( i = j ) with flux on a different three-sphere. Thisthen realises a G geometric transition . However, we will not attempt to relate this toa “large N duality” as in the original discussion in [16]. More concretely, we find six distinct solutions connecting the three classical branches X i , that we will denote X ij .In contrast to the conifold case, we can go from X i with branes to X j with flux or,conversely, from X j with branes to X i with flux, hence X ij = X ji . A rather differentconnection of the three classical branches was discussed in [25], where it was relatedto quantum effects in M-theory. Our discussion, on the other hand, is purely classicaland ten-dimensional.Starting from these relatively simple solutions, we will construct Type IIA solutionswith D2 brane charge and RR C field. This can be done by applying a simple trans-formation analogous to the one discussed in [14] and further studied in [29, 30]. Thesolutions that we obtain in this way have a warp factor (in the string frame metric) thatbecomes constant at infinity, thus the geometry merges into an ordinary G holonomymanifold. By taking a scaling limit we obtain solutions that become asymptoticallyAdS × S × S , albeit only in the string frame. If we further tune the non-trivialparameter, we recover the backgrounds of [31], corresponding to D2 branes and frac-tional NS5 branes transverse to the G manifold S × R . Thus, our solutions maybe thought of as one-parameter deformations of the latter and are analogous to the3aryonic branch deformation [11] of the Klebanov-Strassler geometry [8]. It is thereforevery tempting to think that there should be a close relation between the supersym-metric Chern-Simons theory discussed in [23] and the three-dimensional field theoryon the D2 branes. We will make some speculative considerations in the final section,but we will leave the field theory dual interpretation of our solutions for future work. G holonomy manifold S × R G holonomy manifold S × R that willbe relevant for our discussion. We follow closely the presentation in [25]. The non-compact seven-dimensional manifold defined by X = { x + x + x + x − y − y − y − y = ǫ , x i , y i , ǫ ∈ R } (2.1)is the spin bundle over S and is topologically equivalent to the manifold S × R . For ǫ > S corresponds to the locus y i = 0 and the coordinates y i parameterise thenormal R directions. This manifold admits a Ricci-flat metric with G holonomy, thatat infinity approaches the cone metric ds = dt + t ds ( Y ) (2.2)where Y ∼ = S × S . The Einstein metric ds ( Y ) is not the product of round metricson the two three-spheres. It is in fact a nearly K¨ahler metric, which may be describedin terms of SU (2) group elements as follows [25]. Consider three elements a i ∈ SU (2)obeying the constraint a a a = 1 . (2.3)There is an SU (2) action preserving this relation given by a i → u i +1 a i u − i − , with u i ∈ SU (2), where the index i is defined mod 3. There is also an action by a “triality”group Σ , which is isomorphic to the group of permutations of three elements. This isan outer automorphism of the group SU (2) and may be generated by σ : ( a , a , a ) → ( a − , a − , a − ) ,σ : ( a , a , a ) → ( a , a , a ) . (2.4) In the notation of [25] the generators of the group Σ are denoted as σ = α and σ = β . = { e = σ , σ , σ , σ , σ , σ } , withactions on a i following from (2.4).There are three different seven-manifolds X , X , X , all homeomorphic to S × R ,which can be obtained smoothing out the cone singularity by blowing up three differentthree-spheres inside Y . These are permuted among each other by the action of Σ .This can be seen from the description of the base Y ∼ = S × S in terms of triples ofgroup elements ( g , g , g ) ∈ SU (2) subject to an equivalence relation g i ∼ = g i h with h ∈ SU (2), and is related to the previous description by setting a i = g i +1 g − i − . We canconsider three different compact seven-manifolds X ′ i , bounded by Y , obtained in eachcase by allowing g i to take values in the four-ball B . The non-compact seven-manifoldsobtained after omitting the boundary are precisely the X i . By setting h = g − i − , we seethat each X i has topology S × R , where S and R are parameterised by g i +1 and g i ,respectively. We will review explicit metrics with G holonomy in the three differentcases in the next subsection.We can define three sub-manifolds of Y as C i = { a i = 1 } ∼ = S (2.5)which also extend to sub-manifolds in X i , defined at some constant t . These aretopologically three-spheres, but as cycles in Y and X i they are not independent sincethe third Betti numbers of these manifolds are b ( Y ) = 2 and b ( X i ) = 1, respectively.In fact, we have the following homology relation[ C ] + [ C ] + [ C ] = 0 in Y. (2.6)As cycles in X i , the [ C i ] must obey an additional relation, which in view of theirconstruction above is simply given by [ C i ] = 0 in X i . Therefore the third homologygroup H ( X i ; Z ) is generated by C i − or C i +1 , with [ C i − ] = − [ C i +1 ]. G holonomy metrics A seven-dimensional manifold is said to be a G holonomy manifold if the holonomygroup of the Levi-Civita connection ∇ is contained in G ⊂ SO (7). It is well-knownthat these are characterised by the existence of a G invariant three-form φ (associativethree-form), together with its Hodge dual ∗ φ , which are both closed:hol( ∇ ) ⊆ G iff dφ = d ∗ φ = 0 . (2.7)5he metric compatible with these is Ricci-flat and there exists a covariantly constantspinor, ∇ η = 0. The G invariant forms can be constructed from the constant spinor asbi-linears φ abc = η T γ abc η , ∗ φ abcd = η T γ abcd η , and the metric is then uniquely determinedby these. More generally, the two invariant forms define a G structure on the seven-dimensional manifold. See for example [27].An explicit G holonomy metric on the spin bundle over S was constructed in[20, 21]. In [32] were presented G holonomy metrics on each X i , characterised bythree distinct values of a parameter λ = 0 , ±
1. We will re-derive those results in a waythat will be suitable for a generalisation to be discussed in the next section. We definethe following left-invariant SU (2)-valued one-forms on SU (2) a − da ≡ − i σ i τ i , a da − ≡ − i i τ i , a − da ≡ − i γ i τ i , (2.8)where τ i are the Pauli matrices. We can “solve” the constraint (2.3) by introducingtwo sets of angular variables parameterising the first two SU (2) factors. Then moreexplicitly we have σ + iσ = e − iψ ( dθ + i sin θ dφ ) , σ = dψ + cos θ dφ , Σ + i Σ = e − iψ ( dθ + i sin θ dφ ) , Σ = dψ + cos θ dφ , (2.9)obeying dσ = − σ ∧ σ , d Σ = − Σ ∧ Σ and cyclic permutations. We also have γ i = M ij (Σ i − σ i ), where M ij is an SO (3) matrix. See Appendix A for more details.Introducing the notation da ≡ X i =1 σ i , da ≡ X i =1 Σ i , da ≡ X i =1 (Σ i − σ i ) , (2.10)we can write a metric ansatz in a manifestly Σ covariant form as ds = dt + f da + f da + f da , (2.11)where f i ( t ) are three functions. In order to write the G forms compatible with thismetric, it is convenient to pass to a different set of functions a, b, ω , defined by f = a b ω − , f = b − ω ) , f = b ω ) . (2.12)In terms of these the metric then reads ds = dt + a X i =1 σ i + b X i =1 (cid:0) Σ i − (1 + ω ) σ i (cid:1) (2.13) The one-form Σ should not be confused with the triality group Σ . e t = dt , ˜ e a = aσ a , e a = b (Σ a − (1 + ω ) σ a ) , (2.14)where a is a tangent space index. The associative three-form φ may be convenientlywritten in terms of an auxiliary SU (3) structure as follows φ = e t ∧ J + Re[ e iθ Ω] . (2.15)Here θ is a phase that needs not be constant and the differential forms J, Ω define the SU (3) structure. In terms of the local frame they read J = X a =1 e a ∧ ˜ e a , Ω = ( e + i ˜ e ) ∧ ( e + i ˜ e ) ∧ ( e + i ˜ e ) . (2.16)We can also rewrite φ = e t ∧ J + cos θ Re[Ω] − sin θ Im[Ω] , ∗ φ = 12 J ∧ J + (sin θ Re[Ω] + cos θ Im[Ω]) ∧ e t . (2.17)Imposing dφ = d ∗ φ = 0 gives the following system of first order differential equations Zf ′ = f f √ f + f ) ,Zf ′ = f f √ f + f ) ,Zf ′ = f f √ f + f ) , (2.18)where the prime denotes derivative with respect to t and Z ≡ f f + f f + f f p ( f + f )( f + f )( f + f ) . (2.19)This system can be integrated explicitly in terms of three constants c i defined as c = ( f − f ) ( f + f )( f + f )( f + f ) ( f f + f f + f f ) (2.20)and c , c obtained by cyclic permutations of this expression. Although a priori thereare three independent integration constants, at least one of them must vanish, and theother two are then equal. For example, assuming that c = 0, then f = f and r ≡ c = c = ( f − f ) ( f + f . (2.21)7fter a change of radial coordinate the metric can be written in the form [25] ds ( X ) = dr − ( r /r ) + r (cid:0) − ( r /r ) (cid:1)(cid:0) da − da + 2 da (cid:1) + r da . (2.22)This is a G holonomy metric on the manifold X , where the three-sphere C shrinks tozero at the origin and the three-sphere “at the centre” is homologous to C or − C . Inthe notation of [32] this corresponds to the metric with λ = 0. This metric is invariantunder σ , or equivalently, under the interchange of f ↔ f . In fact, we can re-definethe triality symmetry Σ as acting on the functions f , f , f in the obvious way . Theother two solutions may be obtained for example by acting with the cyclic permutation σ . In the notation of [32] the metric on X corresponds to λ = − X to λ = 1. Notice that the phase θ that enters in the definition of the G structurein (2.15) is not constant, and in particular we havecos θ = λ √ r r / (4 r − r ) / (2.23)with λ = 0 , ±
1, respectively. The conical metric ds = dr + r (cid:0) da + da + da (cid:1) (2.24)may be obtained from any of these by setting r = 0 and is invariant under Σ . Theparameter space of these metrics is depicted in Figure 1.In the figure, moving along an axis corresponds to changing the radius of the three-sphere at the centre of one of the X i spaces, whose volume is √ π r . Hence, inanalogy with the deformed conifold, r measures the amount of “deformation” of theconical singularity. However, in analogy with the resolved conifold, we can also definea parameter measuring the amount of “resolution” of a space X i . Recall that in theresolved conifold the resolution parameter may be defined as the difference of volumesof two two-spheres at large distances [14]. In particular, this parameter measuresthe breaking of a Z symmetry of the singular (and deformed) conifold, consisting inswapping these two-spheres. Here we can define a triplet of resolution parameters, eachmeasuring the breaking of the Z symmetry given by reflection about the three axis inFigure 1. Following [25], we first consider the “volume defects” of the sub-manifolds C ∞ i defined at a large constant value of r . In terms of the radial coordinate t we havethe following asymptotic form of the G metrics ds ( X i ) = dt + t (cid:20) da + da + da − r t (cid:0) ℓ da + ℓ da + ℓ da (cid:1) + O ( r /t ) (cid:21) (2.25) For the elements of order two, we must also reverse the orientation of the seven-dimensional space. G holonomy metrics on S × R has three branches permuted by the action of the group Σ . The three G holonomymetrics are invariant under elements of order two σ ij , which are reflections about eachof the three axis. The intersection point of the three branches corresponds to thesingular G cone. In the notation of [32]: X corresponds to λ = 0, X corresponds to λ = −
1, and X corresponds to λ = 1.where t = r − r / (4 r ) + O ( r − ) and the constants ( ℓ , ℓ , ℓ ) take the values: ( − , , X , (1 , − ,
1) for X , and (1 , , −
2) for X . Then for the “volume defects” we havevol( C ∞ i ) = 1627 π t + 29 π r ℓ i (2.26)and we may define the i th resolution parameter as α res i ≡ vol( C ∞ i +1 ) − vol( C ∞ i − ) = 29 π r ( ℓ i +1 − ℓ i − ) . (2.27)To see that this is a sensible definition, let us evaluate α res1 in the three cases X i . Wehave α res1 ( X ) = 0 , α res1 ( X ) = − π r , α res1 ( X ) = + 23 π r . (2.28)The interpretation is that the manifold X preserves the Z reflection about the axis1, hence from this point of view, r is a “deformation” parameter. On the other hand,the manifolds X and X break this symmetry in opposite directions. From the pointof view of X , X is a “resolution” and X its flopped version. Notice in particularthat we cannot have “resolution” and “deformation” at the same time, exactly as ithappens for the conifold in six dimensions. Indeed, the relation to the conifold maybe made very precise by considering the different G holonomy metrics (times R , )as solutions of M-theory. Then there exist three different reductions to Type IIA that9ive rise to manifolds with topologies of the deformed, resolved, and flopped resolvedconifold, respectively [25, 26].Finally, let us recall some facts about the cohomologies of these spaces. For each X i the third cohomology group is H ( X i ; Z ) = Z , so there is only one generator, that canbe chosen to integrate to one on the non-trivial three-cycle. However, it is convenientto introduce the following set of three-forms η = − π σ ∧ σ ∧ σ , η = 116 π Σ ∧ Σ ∧ Σ , η = − π γ ∧ γ ∧ γ , (2.29)which are exchanged by the action of Σ . We also have that η = − π (Σ − σ ) ∧ (Σ − σ ) ∧ (Σ − σ ) . (2.30)Integrating the η j over the sub-manifolds C i , we have the relation Z C i η j = δ j,i +1 − δ j,i − (2.31)and by Poincar´e duality η i → C i the intersection numbers C i · C j = δ j,i +1 − δ j,i − . Noticethat having fixed the orientation so that C · C = +1, we then have that C · C = +1,which gives R C σ ∧ σ ∧ σ = − π so that the orientation of C is opposite to thatof C . S × R G holonomy manifold X i . The backreaction of the fivebranes modifies thegeometry, making the internal space a smooth torsional G manifold . As we will see,the topology is again R × S , although we will be careful about which S . Since we areinterested in solutions arising from NS fivebranes, we may work in Type I supergravity,and allow for a non-trivial three-form H and dilaton profile. By applying an S-dualityto the NS5 branes in Type IIB, these solutions may also be interseted as arising fromD5 branes. G solutions General classes of supersymmetric solutions of Type I and heterotic supergravities havebeen studied in [27], extending the works of [17] and [18]. Here we are interested in10olutions where the non-trivial geometry is seven-dimensional and is characterised bya G structure. We will therefore refer to this class as torsional G solutions . Theten-dimensional metric in string frame is unwarped ds str = dx + ds . (3.1)The supersymmetry equations are equivalent to exterior differential equations obeyedby the G structure on the seven-dimensional space with metric ds and read [27, 28]: φ ∧ dφ = 0 d ( e − ∗ φ ) = 0 e ∗ d ( e − φ ) = − H (3.2)where Φ is the dilaton field and ∗ denotes the Hodge star operator with respect tothe metric ds . The Bianchi identity dH = 0 implies that all remaining equations ofmotion are satisfied. See [27] for a more detailed discussion of this type of G structure.Examples of solutions to these equations were discussed in [33, 23, 27, 22] and we shallreturn to some of these later. We will now specify ansatze for the metric, G structure and H field. Although the G structure determines the metric uniquely, and the H field is then derived fromthe third equation in (3.2), we find more convenient to start with an ansatz for H that is manifestly closed dH = 0. Specifically, we use the ansatz for the metric andassociative three-form discussed earlier ds = M (cid:2) dt + f da + f da + f da (cid:3) φ = M / (cid:2) e t ∧ J + Re[ e iθ Ω] (cid:3) (3.3)where we inserted a factor of M in front of the metric. For the three-form flux we take H = 2 π M (cid:20) γ η + γ η + γ η + γ dt ∧ X i =1 σ i ∧ Σ i (cid:21) (3.4)where the factor of 2 π M is again for convenience. Imposing dH = 0 implies γ i = γ + α i i = 1 , , , γ = γ ′ π , (3.5)where γ is a function and α i are three integration constants. The ansatz then dependson four functions f i , γ and three constants α i , although we will see that the homology11elation among the C i implies that only two of these are significant, and are fixed byflux quantisation and regularity of the metric.The action of the Σ symmetry on the functions in the ansatz is given by σ : ( f , f , f ) → ( f , f , f ) σ : ( f , f , f ) → ( f , f , f ) σ : ( γ , γ , γ ) → ( − γ , − γ , − γ ) σ : ( γ , γ , γ ) → ( γ , γ , γ ) (3.6)with the rest following from group multiplication rules. The minus signs in the action ofthe order two elements on the γ i arise because the orientation of the seven-dimensionalspace is reversed, hence the Hodge ∗ operator in (3.2) changes sign.Inserting the ansatz into the equations (3.2), after some computations, we arrive ata system of first order ODEs. We relegated some details in Appendix B. We havefour coupled ODEs for the functions f i , γ , while an additional decoupled equationdetermines the dilaton profile in terms of the other functions. Although the explicitform of the equations is rather complicated, its presentation can be simplified slightlyby organising it in terms of the Σ symmetry. We have D ( f i , γ i ) f ′ = F ( f , f , f , γ , γ , γ ) D ( f i , γ i ) f ′ = F ( f , f , f , − γ , − γ , − γ ) D ( f i , γ i ) f ′ = F ( f , f , f , − γ , − γ , − γ ) D ( f i , γ i ) γ ′ = G ( f i , γ i ) (3.7)where, defining Q ≡ f f + f f + f f , the functions F , G and D are given by F ( f , f , f , γ , γ , γ ) = 768 f f ( f + f )( f + f ) + γ ( f + f ) + γ γ (3 f − Q )+ γ γ ( Q + 2 f f − f ) + γ γ ( Q + 2 f f − f )+ 32 γ Q ( f − f ) − γ Q ( f + f ) + 32 γ Q ( f + f ) ,G ( f i , γ i ) = − (cid:2) γ ( f + f )( Q + f f ) + γ ( f + f )( Q + f f ) , (3.8)and √ D ( f i , γ i ) = 32 (cid:0) γ ( f + f ) + γ ( f + f ) + γ ( f + f ) + 2 γ γ f (3 Q − f ) + 2 γ γ f (3 Q − f ) + 2 γ γ f (3 Q − f )+ 96 Q (cid:0) γ ( f − f ) + γ ( f − f ) + γ ( f − f ) (cid:1) + 2304 Q ( f + f )( f + f )( f + f ) (cid:1) / Q / . (3.9)The decoupled equation for the dilaton reads2 √ QD ( f i , γ i ) Φ ′ = P ( f i , γ i ) (3.10)12here P ( f i , γ i ) = 2 γ ( f + f ) + 2 γ ( f + f ) + 2 γ ( f + f ) + 4 γ γ f (3 Q − f ) + 4 γ γ f (3 Q − f ) + 4 γ γ f (3 Q − f )+ 96 Q (cid:0) γ ( f − f ) + γ ( f − f ) + γ ( f − f ) (cid:1) . (3.11)Once a solution for f i , γ is determined (for example numerically), then the dilaton canbe obtained integrating (3.10). Notice that D and P are invariant under Σ and G isinvariant up to an overall change of sign under transformations of order two elements.It follows that Φ is invariant under Σ . The phase θ in the associative three-form is anon-trivial function of f i , γ i , whose explicit form can be found in Appendix B.From the BPS system it is clear that generically, for any given solution we havein fact six different solutions, obtained acting with Σ . To study the system we cantherefore focus on one particular case. Notice that if we formally set γ i = 0 in (3.7),then we recover the G holonomy BPS equations (2.18). Solutions to this system werepresented in [33], [23, 24] and [22]. In particular, the solution of [23] corresponds tothe near brane limit of a configuration of M NS fivebranes wrapped on an S insidethe G manifold S × R . Below we will be more precise about which G manifold X i is relevant for a particular solution of the type discussed in [23]. Maldacena-Nastase solutions
The basic solution of [23] may be recovered from our general ansatz by setting f + f = 1 / ,α = α = − α = − . (3.12)For consistency these conditions impose also γ = γ − − f . Then we are leftwith two unknown functions, f and γ . To make contact with the variables in [23] onehas to set f = 4 R + γ −
132 (3.13)and then passing to the variables a, b, ω we have a = R , b = 14 , ω = γ , (3.14)so that the metric reduces to the ansatz in [23], namely ds = M " dt + R X i =1 σ i + 14 X i =1 (cid:0) Σ i − (1 + ω ) σ i (cid:1) . (3.15) Our one-forms are related to those in [23] as: σ here i = w there i and Σ here i = ˜ w there i . H reduces to that in [23]. In terms of the functions R and ω , the system(3.7) reduces to the system in the Appendix of [23]. As discussed in [23, 24] thereexists a unique non-singular solution to the differential equations. In the interior thetopology of the solution is S × R , where the three-sphere C smoothly shrinks to zeroand the three-cycle is represented by C or − C . Then, more precisely, the topology ofthis solution is that of the G manifold X . The authors of [23] discussed also a secondsolution which can be obtained from the basic solution by acting with σ . Hence thetopology of this solution is that of G manifold X . It is clear that there are four moredifferent solutions, obtained by acting with elements of Σ . Finding analytic solutions to the BPS system (3.7) seems very difficult. As usual inthese cases, we will then turn to a combination of numerical methods and asymptoticexpansions. We are interested in non-singular solutions to the system, giving riseto spaces with topology S × R . As for the G holonomy manifolds X i and theMaldacena-Nastase solutions, we then require that two functions f i go to zero in theinterior, while the third function approaches a constant value, parameterising the sizeof the non-trivial S inside S × R . We can restrict our attention to one particularcase, for example we may require that f and f go to zero in the IR (at t = 0) while f approaches a constant value f (0) ≡ c >
0. This then has the topology of X , where C shrinks to zero. This solution was studied in [22].More generally, we impose boundary conditions such that the topology of the solutionis that of one of the manifolds X i . This fixes the values of the integration constants α i . Using the relation (2.31) we can evaluate the flux of the three-form H (3.4) on thesub-manifolds C i , defined exactly like in (2.5), and at some constant t . We have q i ≡ π Z C i H = M γ i +1 − γ i − ) = M α i +1 − α i − ) (3.16)where the result does not depend on t . The q i then obey the relation q + q + q = 0,reflecting the homology relation [ C ] + [ C ] + [ C ] = 0. Hence we can parameterise theconstants α i by taking for example α = − k , α = − k , α = k , (3.17) Asymptotically the geometry is a linear dilaton background. Notice that as t → f → f → f → /
14o that q = − M k , q = M k + k ) , q = M − k + k ) . (3.18)The constants k , k are determined for a given solution as follows. Suppose that werequire the manifold to have the topology of X . Then the flux of H through C isminus the flux through C , namely q = − q . In terms of the constants k , k , then wemust have k = k = k and k can be reabsorbed in the definition of M . There are thenessentially two choices for k , namely k = ±
1, corresponding to two different solutions,both with topology of X . We denote these two solutions as X and X , respectively.More generally, there are six different solutions and we denote the corresponding spacesas X ij . The topology of the spaces X ij is the same as the G holonomy manifolds X i .In each case the flux through the non-trivial cycle must be quantised, thus we requirethat N ( X ij ) = | ǫ ijk | π Z C k H = M . (3.19)The signs have been chosen to give always a positive number and are consistent withthe action of Σ . In conclusion, flux quantisation, together with the condition thatthe flux through the vanishing three-sphere vanishes, fixes the integration constants k , k in all cases. We summarise the values of the k , k and the q i in each of the sixsolutions in Table 1. k k q q q X − M M X − − M MX − − M MX − − M − M X − M − MX M − M Table 1: Values of k , k and q i for the six different solutions X ij . The basic Maldacena-Nastase solution is the c = 1 / X , while the second Maldacena-Nastasesolution is the c = 1 / X . For definiteness, let us concentrate on the case of X . To discuss the expansions itis convenient to first rescale the radial coordinate by a constant factor as t → √ ct .15he boundary conditions that we impose at t = 0 determine the expansions of thefunctions f i and γ around t = 0 as follows f + f = 18 ct + 1 − c c t + O (cid:0) t (cid:1) f − f = 196 t + 3 − c c t + O (cid:0) t (cid:1) f = c + − c c t + − c + 2048 c c t + O (cid:0) t (cid:1) γ = 1 − t + − c c t + O (cid:0) t (cid:1) (3.20)The corresponding expansion for the dilaton readsΦ = Φ − c t + −
293 + 21504 c c t + O (cid:0) t (cid:1) (3.21)where Φ is an (IR) integration constant. We therefore have a family of non-singularsolutions, parameterised by the constant c , measuring the size of the non-trivial S .Using numerical methods one can then check that, for any value of c ≥ /
8, thereexists a non-singular solution approaching (3.20) as t →
0. The special value c =1 / X (in the interior), generalising thesolution discussed in [23]. Towards infinity we find two different types of asymptotic expansions of the functions.In one expansion the functions have the following behaviour for large t : f = ct
36 + 14 − ct + O (cid:0) t − (cid:1) f = ct − − ct + O (cid:0) t − (cid:1) f = ct
36 + 6916 ct + O (cid:0) t − (cid:1) γ = 13 + O (cid:0) t − (cid:1) Φ = Φ ∞ + O (cid:0) t − (cid:1) (3.22)where Φ ∞ is an (UV) integration constant. Notice that the constant c appears heretrivially because of the rescaling t → √ ct we made, therefore at this order we don’t see If we had left the constants k , k arbitrary, the IR expansions of the functions f i , γ in powerseries, would impose γ (0) = k = k , that is q = 0.
16 genuine UV integration constant. After this particular order the expansion in inversepowers of t is not valid anymore and one would need to use other types of series togain more precision. This expansion can be matched numerically to the IR expansionsfor all values of c > / f i all have the same leading behaviour in t towards infinity, corresponding to the G holonomy cone. From the sub-leading terms we can also read off an effective“resolution parameter”, measuring the amount of Z symmetry breaking in each case.The asymptotic form of the metric here is ds ( X ij ) = M c (cid:20) dt + t
36 ( da + da + da ) + 14 c (cid:0) ℓ da + ℓ da + ℓ da (cid:1) + O (1 /t ) (cid:21) (3.23)where ( ℓ , ℓ , ℓ ) = (1 , − ,
0) for X and the remaining ones are determined by the Σ action. The “volume defects” are given byvol( C ∞ i ) = ( M c ) / (cid:18) π t − π ℓ i tc (cid:19) . (3.24)Notice that even after subtracting the leading divergent part these volumes are now“running”. This running is analogous to the running volume of the two-sphere atinfinity in the resolved deformed conifold, although here the running is linear, ratherthan logarithmic. Then the i th effective resolution parameter may be defined as α res i ≡ vol( C ∞ i +1 ) − vol( C ∞ i − ) = ( M c ) / π ( ℓ i − − ℓ i +1 ) tc . (3.25)In Section 2.2 we saw that in the G holonomy manifold X i the resolution parameter α res i was vanishing, reflecting a Z ⊂ Σ symmetry of the geometry. Hence, the relevant resolution parameter to consider for the manifolds X ij is α res i . For example, we findthat α res3 ( X ) = − α res3 ( X ) = 8 π ( M c ) / tc , (3.26)which is again running. Notice that keeping M c fixed, a non-zero value of the parameter c − may then be interpreted as turning on a “resolution” in the manifold X . Indeed,we will show below that the limit c → ∞ gives the G holonomy manifold X . Of course α res i +1 and α res i − are also non-zero, since the solutions do not preserve any Z symmetry.However these are less interesting parameters, since they are non-zero also in the G holonomy manifold X i .
17e also find a second type of expansion at large t , in which the behaviour of thefunctions is different and we have f = √ t + κ − √ t + 1 − κ t + 7 − κ + 512 κ √ t + O (cid:0) t − (cid:1) γ = √ t + 1 − κt + 9 − κ + 256 κ √ t + O (cid:0) t − (cid:1) Φ = − √ t + 38 log t + Φ ∞ + 3(1 + 32 κ )16 √ t + 29 − κ − κ t + 9 − κ + 3072 κ + 32768 κ √ t + O (cid:0) t − (cid:1) (3.27)In this case the expansions remain valid at high orders, hence presumably they don’tbreak down. These expansions may be matched numerically to the IR expansions forthe particular case c = 1 /
8, thus they correspond to the Maldacena-Nastase solution .Despite the fact that κ seems a free constant, it can be determined numerically to be κ ≈ − . As can be seen from the expansions, while the behaviour of the functions in the IRchanges smoothly as we vary the parameter c , the behaviour in the UV changes dis-continuously if we choose the extremal value for the parameter c = 1 /
8. Here wepresent some plots of the numerical solutions to illustrate the qualitative behaviour ofthe metric functions f i for various values of c .In Figure 2 we show plots of the functions for large values of c . We see that in the IRthe behaviour is that of the space X . However, despite starting below f (at zero), thefunction f eventually crosses f , in agreement with the UV expansions. The crossingpoint moves further and further along the radial direction as c is increased. In Figure3 we plot the functions for values of c close to the minimum. We see that in the IRthe functions are all very close to the special case c = 1 /
8. However, when c is notexactly equal to its minimum value, the functions start to deviate at some point. Forvalues of c closer and closer to the special one, there is a larger and larger region wherethe functions are well approximated by the profiles of the Maldacena-Nastase solution. The factor of √ t → √ ct . This behaviour is analogous to that of the one-parameter family of solutions discussed in [11, 14].In that case, for the special value of the integration constant γ = 1, one obtains the solution of [9],which has linear dilaton asymptotics. Figure 2: Plots of the functions c − f i for the X solution for different values of c . f isin red, f in purple and f in blue. The factor c − is there for normalisation purposes.From the top left to the bottom right, the values of c are increasing and are 0.2, 0.3, 0.4for the first three. The bottom-right plot corresponds to the space X where f = f .This is formally the plot for c → ∞ .Finally, in Figure 4, we plotted the dilaton and the function γ for various values of theconstant c . We see that the generically e Φ goes to a constant e Φ ∞ at infinity, while inthe particular case of Maldacena-Nastase e Φ vanishes in the UV. In this subsection we analyse two special limits of the one-parameter solutions, namely c → ∞ and c ∼ /
8, respectively.
The solution for c → ∞ : G holonomy with flux The numerical solutions show that by increasing the value of c the solution X looksmore and more like the G manifold X . To see this more precisely, we will consider19
10 15 200.511.522.533.54 5 10 15 200.511.522.533.545 10 15 200.511.522.533.54 5 10 15 200.511.522.533.54
Figure 3: Plots of the functions c − f i for the X solution for different values of c . f is in red, f in purple and f in blue. The factor c − is there for normalisationpurposes. From the top left to the bottom right, the values of c are increasing and are0 .
125 (the exact minimum value), 0.125001, 0.12501 and 0.126. The first plot is theMaldacena-Nastase solution.
Figure 4: On the left are plots of the function e Φ − Φ for different values of c . Onthe right, plots of the function γ . The orange plots correspond to the minimum value c = 0 . c = 0 . c = 0 .
15 and the blue ones to c = 0 .
5. In the Maldacena-Nastase solution at infinity there is a linear dilaton and the H vanishes.an expansion of the functions f i and γ in inverse powers of c of the form: f i = c ∞ X n =0 c n f i ( n ) ,γ = ∞ X n =0 c n γ ( n ) . (3.28) From (3.29) it can be checked a posteriori that the first few terms reproduce the UV expansions(3.22). Therefore the series (3.28) is certainly not valid for c = 1 /
8. In any case, we only need this t it can be checked that this agrees with the IR expansions (3.20). Then onecan solve the system (3.7) order by order in powers of c − . There are of course differentsolutions depending on the boundary conditions and here we will concentrate on theboundary conditions already treated in Section 3.3. For our purpose, we only need thefirst few orders of the expansion (3.28). These read f = f = r − r r , f = 2 r + r r ,f = − f = − r + r r + r r − r r ( r + r r + r ) , f = 0 ,γ (0) = r + r r + r r + 6 r r ( r + r r + r ) , (3.29)where r is a function of the radial coordinate t defined as in Section 2.2, namely drdt = r − r r . (3.30)The functions f i (0) at the lowest order in the expansion solve a simplified version of(3.7) where γ i = 0 , f = f , which are simply the differential equations for the G holonomy metric X . The metric in terms of the expansion (3.28) reads ds = M c (cid:2) ds ( X ) + O (cid:0) c − (cid:1)(cid:3) (3.31)thus, at leading order in c , the solution looks like the G manifold X with a very large S , and M units of H flux through it. One could of course take c → ∞ while keeping M c fixed, by taking M → H vanishes. The solution for c ∼ / : G holonomy with branes Here we show that when c is very close to the minimum value c = 1 /
8, there is aregion where the solution X ij looks like a G holonomy manifold X j with M fivebraneswrapped on the non-trivial three-sphere. The calculation is analogous to that appearingin Section A.2 of [14].The solution stays close to the Maldacena-Nastase solution up to large values of t . To analyse the behaviour of the solution where it starts departing from this, we for large c here. f = √ t + µ , f = 116 + µ ,f = 116 + µ , γ = √ t . (3.32)The leading terms are those of the Maldacena-Nastase solution and we require that µ ≪ t , µ , µ ≪
1. Anticipating the form of the metric that we are after, we changecoordinates as follows ds = M (cid:20) f (cid:18) dy + da (cid:19) + f da + f da (cid:21) . (3.33)We could also have taken f in front, but since we will find that µ = µ , this doesnot matter. Then we plug the ansatz (3.32) into the BPS equations (3.7) and expandto first order in the µ i . The equation for γ ′ is satisfied automatically at leading order.Working at large y , we can solve the equations for µ i and we find µ = β e √ y/ −
1) + β ,µ = µ = β e √ y/ , (3.34)where β , β are two integration constants. We can determine the dilaton with thesame precision by considering the ansatzΦ = − √ t + µ (3.35)where µ ≪ t . Then we find µ = 8 β ( e √ y/ −
1) + Φ (3.36)where Φ is another integration constant. Inserting these back into the metric andchanging coordinates as r = 4 √ M β e √ y/ we find ds ≈ (cid:18) Mr (cid:19) " dr + r X i =1 (cid:0) Σ i + (Σ i − σ i ) (cid:1) + M √ y X i =1 σ i ,e − Φ ) ≈ β e − β (cid:18) Mr (cid:19) . (3.37)This is the approximate solution for M fivebranes in flat space, wrapped on the three-sphere parameterised by σ i . More precisely, we see that the topology is that of S × , where the three-sphere C transverse to the branes (defined by σ i = 0) vanishessmoothly. Hence this is the same topology of the G holonomy manifold X . Thefivebranes then can be wrapping C or − C .This approximation however requires that y is large, but at the same time y ≪ y , where y is defined by β = e −√ y / . Presumably around y ∼ y the solutionlooks more accurately like X [14], but this seems difficult to analyse in the linearisedapproximation. We can also estimate the relation between c and y by extrapolatingto zero the value of f + f . This gives c − / ≈ e − √ y . (3.38) In this section we have discussed a set of gravity solutions characterised by a non-trivialparameter c . The additional parameters of the solutions are the M integral units ofNS three-form flux H and the asymptotic value of the dilaton Φ ∞ . The constant Φ is a function of Φ ∞ and c , that may be determined numerically. There are six differentsolutions, exchanged by the action of the triality group Σ . In each case the internalseven-dimensional manifold is an asymptotically conical space, with topology S × R ,that we have denoted X ij , with i, j = 1 , ,
3. The base of the asymptotic cone is thenearly K¨ahler manifold S × S with metric ds ( Y ) = ( da + da + da ) [25]. Moreprecisely, the topology of the space X ij is that of the G holonomy manifold X i , thatwe reviewed in Section 2.In each case the parameter c gives the size of the non-trivial S at the origin, hencethis is analogous to the deformation in the deformed conifold. On the other hand, onecan also define a resolution parameter by looking at how the metric breaks a Z ⊂ Σ symmetry at large distances. In particular, we have argued that the parameter 1 /c gives a measure of how much the space X ij deviates from the X i geometry. Hence,from this point of view, 1 /c can be interpreted as an effective resolution parameter.In the case of G holonomy the moduli space of metrics on S × R has three differentbranches, meeting at the origin. With an abuse of language , we can say that thesingular G cone over S × S may be deformed, resolved or flopped-resolved, with thethree possibilities mutually exclusive. The six solutions that we discussed may be saidto be deformed and resolved, analogously to the resolved deformed conifold geometry[11, 14]. In particular, we do not use these words here in the sense of complex or symplectic geometry. c is very large the solution approaches a G holonomy manifold with flux on alarge three-sphere. When c hits the lower bound c = 1 /
8, the X ij geometry becomes asolution of the type discussed by [23], corresponding to the near-brane limit of a largenumber of fivebranes wrapped on the S inside a G manifold with topology S × R .These have a finite size three-sphere at the origin, but are asymptotically linear dilatonbackgrounds. When c is very close to the critical value c = 1 / t (see the plotsin Figure 3) and when it starts deviating from this behaviour the geometry becomesapproximately that of the G manifold X j with M fivebranes wrapping the non-trivialthree-sphere inside this.Figure 5: On the left: the solutions X and X interpolate continuously between the G manifold X with flux and the G manifolds X and X with branes, respectively.The two solutions are related by the Z symmetry σ and both have topology of X ∼ = S × R . On the right: the solution X interpolates continuously between the G manifold X with flux and the G manifold X with branes, while the solution X interpolates between the G manifold X with flux and the G manifold X withbranes. The two solutions are related by the same Z symmetry σ , however X hasthe topology of X ∼ = S × R while X has the topology of X ∼ = S × R .For each X ij solution the parameter c interpolates between the G holonomy manifold X i with M units of flux on a large three-sphere, and the G holonomy manifold X j with M fivebranes wrapped on a (different) three-sphere. Hence, this may be interpreted asa realisation of a G geometric transition, purely in the context of supergravity. Noticethat this is different from the closely related setup in [26, 25], where the relevantgeometric transition involved D6 branes wrapped in the conifold, although this wasembedded in the G holonomy context by uplifting to M-theory.24rom the point of view of the G holonomy manifold X (say) with M units offlux through the three-sphere C ∼ = − C , the two solutions X and X break the Z symmetry under σ in two opposite directions. These look like a “resolution” of themanifold X and its flopped version. The breaking of this Z symmetry is analogousto the breaking of the Z symmetry in the resolved deformed conifold. On the otherhand, from the point of view of the branes wrapped on the three-sphere C ∼ = − C in X , the two solutions X and X break the Z symmetry generated by σ by“deforming” the original X manifold in two different ways. In other words, the G geometric transition may proceed from branes in X to flux in X or from branes in X to flux in X . There is no analogue of this in the conifold case.Moreover, depending on which three-sphere of X the branes wrap, in the geometryafter the backreaction this three-sphere may become contractible or not. For example,if the fivebranes were wrapped on C ⊂ X and after the geometric transition we havethe manifold X (with flux), the sphere wrapped by the branes is contractible. Whereasif the fivebranes were wrapped on C ⊂ X and after the geometric transition we havethe manifold X (with flux), the sphere wrapped by the branes is not contractible. Thisphenomenon has no analogue in the conifold case, where the two-sphere wrapped bythe branes always becomes contractible in the backreacted geometry. In the context ofthe discussion in [23] these two possibilities led to two different values of the quantitydenoted k , defined as the flux of H through the three-sphere wrapped by the branes.In particular, for the basic solution of [23] (with X topology) it was assumed thatthe three-sphere wrapped by the branes is C ⊂ X and therefore k = q = M . Thesecond solution of [23] (with X topology) was interpreted as arising from fivebranesstill wrapped on C ⊂ X and therefore here k = q = 0. Cf . Table 1. This ambiguitymay also be understood as related to different gauge choices for the connection onthe normal bundle to the wrapped three-sphere. In [23] it was explained how thiscorresponds to changing the number of fermionic zero modes on the brane worldvolumewith all choices leading to equivalent results for the physical Chern-Simons level of thedual gauge theory, namely | k | = M/ G struc-tures In this section we will discuss solutions of Type IIA supergravity of the type R , × w M ,where the internal seven-dimensional manifold M has a G structure and there are25arious fluxes. We will then show that starting from a torsional G geometry onecan obtain a more general solution, interpolating between the original solution and awarped G holonomy solution. The method that we use is quite general and can beapplied to supergravity solutions different from the ones of the previous section. We write the metric ansatz in string frame as ds str = e / (cid:0) dx + ds (cid:1) . (4.1)The solutions are characterised by a G structure on the internal space, namely anassociative three-form φ (and its Hodge dual) and a non-trivial phase ζ . The non-zerofluxes are the RR four-form F and the NS three-form H . The equations characterisingthe geometry may be written in the form of generalised calibration conditions [34], andcan be obtained straightforwardly by reducing the equations presented in [35] fromeleven to ten dimensions. Some details of this reduction are presented in Appendix C.The equations read d (cid:0) e ∗ φ (cid:1) = 0 ,φ ∧ dφ = 0 , dζ − e − cos ζ d (cid:0) e sin ζ (cid:1) = 0 ,d (cid:0) e / cos ζ (cid:1) = 0 . (4.2)In addition, the fluxes are determined as follows H = 1cos ζ e − / ∗ d (cid:0) e cos ζ φ (cid:1) ,F = vol ∧ d (cid:0) e sin ζ (cid:1) + F int4 , F int4 = − sin ζ cos ζ e − d (cid:0) e cos ζ φ (cid:1) . (4.3)Notice the relation sin ζ e ∆ − / H + ∗ F int4 = 0 . (4.4)From the results of [35] we have that any solution to these conditions, supplementedby the Bianchi identities dH = dF = 0, solves also the equations of motion. Thisgeometry is the G analogue of the interpolating SU (3) structure geometry discussedin [29, 30]. Notice that this case is not contained in the equations presented in [36],which instead describe an SU (3) structure in seven dimensions. Although the ansatzfor the bosonic fields in the latter reference is equivalent to ours, the ansatz for the26illing spinors does not allow the structure that we are discussing here. The interestedreader can find a discussion of spinor ansatze in Appendix C.The conditions (4.2), (4.3) include the Type I torsional geometries as a special case,which are obtained simply setting ζ = π . The warp factor is related to the dilaton as e = e − so that the ten-dimensional metric in string frame is unwarped. The limitcos ζ → G structure in eleven dimensionsfrom which our equations have been obtained breaks down . However, going backto the equations in [35], one can see that in this limit the internal eight-dimensionalgeometry becomes a warped Spin(7) manifold with self-dual flux [37]. A careful analysisthen shows that in the cos ζ → G holonomy solutionsderived in [31]. The warp factor is again related to the dilaton e = e − ≡ h (4.5)and rescaling the internal metric as ds = hd ˆ s the full metric becomes ds str = h − / dx + h / d ˆ s (4.6)where the rescaled metric now has G holonomy , namely d ˆ φ = d ˆ ∗ ˆ φ = 0. Takingdirectly the limit on the relation (4.4) gives H + ˆ ∗ F int4 = 0. Hence the four-form fluxcan be written as F = vol ∧ dh − − ˆ ∗ H . (4.7)The equation of motion for F implies that the warp factor is harmonic with respectto the G metric, namely ˆ ✷ h = − H . (4.8) We now discuss two different methods to generate solutions of the equations presentedabove, starting from a solution of the Type I torsional system. One method, analogousto the procedure discussed in [14], involves a simple chain of dualities. Another methodexploits the form of the supersymmetry conditions. We will refer to this second methodas “rotation” [29, 30]. An integration constant can always be reabsorbed in a scaling of the coordinates. In particular, the one-form K does not exist. See Appendix C and [35]. ualities We start with a solution of (3.2). The non-trivial fields are the dilaton Φ,the three-form flux H and the metric ds str = dx + ds . First we uplift to elevendimensions. We rescale the new eleventh dimension by a constant factor e − Φ ∞ , boostalong x with parameter β , and finally undo the rescaling of x . This gives thetransformation t → cosh βt − sinh βe Φ ∞ x , x → − sinh βe − Φ ∞ t + cosh βx . (4.9)Then we reduce back to Type IIA along the transformed x and we perform two T-dualities along the two spatial directions of the R , part. At the level of brane charges,the steps in the transformation may be summarised asNS5 → M5 → M5, p KK → NS5, D0 → NS5, D2Notice that a non-zero magnetic ˆ C field will be generated in the process. The dualitiesabove result in the following Type IIA solution d ˆ s str = h − / dx + h / ds , h = 1 + sinh β (1 − e − − Φ ∞ ) ) , ˆ H = cosh βH , e = e h / , ˆ F = − e − Φ ∞ tanh β vol ∧ d ( h − ) + sinh βe Φ ∞ e − ∗ H , (4.10)where here the hatted quantities denote the new solution while the unhatted onesdenote the initial solution. Notice that in contrast to the case in [14] the dilaton ischanged in the transformation. This can be understood because here the procedureintroduces D2 branes, to which the dilaton couples. Notice also that we need h > e − ∞ > tanh β . Thus the transformation may be applied only if inthe initial solution the dilaton is a bounded function. We can write the transformedfluxes as ˆ H = − cosh βe ∗ d ( e − φ )ˆ F = − e − Φ ∞ tanh β vol ∧ dh − − sinh βe Φ ∞ d ( e − φ ) (4.11)from which we can read off the internal ˆ C field in terms of the associative three-form φ , namely ˆ C = − sinh βe Φ ∞ − φ . (4.12)From these expressions it is clear that the Bianchi identities of the initial solution implythe ones of the transformed solution. In principle this method may be applied also tonon-supersymmetric solutions. 28 otation The same transformation can be done directly on the supersymmetry equa-tions, without doing any dualities. One advantage of this method is for example thatit is applicable to configurations with sources [30]. Suppose that Φ (0) = − (0) and athree-form φ (0) are a solution of the system (3.2). Then one can defineˆ φ = (cid:18) cos ζc (cid:19) φ (0) ,e = cos ζc e (0) ,e = (cid:18) c cos ζ (cid:19) e − Φ (0) , (4.13)and a new seven-dimensional metric d ˆ s = c − cos ζ ds (0)27 . It is easy to check that thenew quantities ˆΦ, ˆ∆ and ˆ φ are a solution of the first three equations of the generalsystem (4.2). The fourth one can be solved and it gives a relation between ζ and thedilaton of the original solution: sin ζ = c e − Φ (0) . (4.14)Here c and c are integration constants. The rotated background in terms of unrotatedquantities reads d ˆ s str = h − / dx + h / ds (0)27 , h = 1 c (cid:16) − c e − (0) (cid:17) , ˆ H = 1 c e (0) ∗ (0)7 d (cid:0) e − (0) φ (0) (cid:1) , e = e (0) h / , ˆ F = 1 c vol ∧ dh − − c c d (cid:0) e − (0) φ (0) (cid:1) . (4.15)In order to match the result of this method to the previous one, one has to identify c = − β , c = − e Φ ∞ tanh β . (4.16)As before, the Bianchi identities of the general solution follow immediately from theones of the unrotated solution. G holonomy solutions We can now apply the transformation above to the solutions of Section 3. Notice thatindeed in those solutions the dilaton was bounded from below. For each solution ofSection 3 we then obtain a one-parameter family of solutions of Type IIA supergravity,29ith D2 brane charge and an internal C field. The background is simply obtained byplugging the solutions of Section 3 into the equations (4.10) or (4.15). Notice that thewarp factor h in (4.10) goes to one at infinity. However, for AdS/CFT applications,one would like to take a decoupling limit in which the warp factor goes to zero atinfinity. In this way the asymptotically Minkowski region is removed and replaced bya boundary. We will be more precise about the asymptotics momentarily. To proceed,first recall that one should quantise the transformed three-form ˆ H as˜ M = 14 π Z S ˆ H = M cosh β ∈ N (4.17)where S is the appropriate non-trivial three-sphere in each case. Then rescaling theMinkowski coordinates as x µ → ˜ M cosh βc ! / x µ , (4.18)in the limit β → ∞ , keeping ˜ M fixed, the metric is finite and reads d ˆ s str = ˜ M h ˜ h − / c − dx + ˜ h / d ˜ s i . (4.19)Here d ˜ s does not have a factor of M and the new warp factor ˜ h = 1 − e − − Φ ∞ ) goesto zero at infinity. The factor of c makes sure that the asymptotic form of the metric isindependent of c and in addition will allow us to take the further limit c → ∞ . Fromthe expressions in (4.10) we see that this limit is problematic for the transformed ˆ F and dilaton ˆΦ. To obtain a finite limit we also send e Φ ∞ →
0, while keeping fixed e ∞ sinh β = c . (4.20)The factor c on the right-hand side is again inserted to allow to take a further c → ∞ limit in the solution. Now taking β → ∞ the solution is then completed with e = c e − Φ ∞ ) ˜ h / , ˆ H = − ˜ M e − Φ ∞ ) ˜ ∗ d ( e − − Φ ∞ ) ˜ φ ) , ˆ F = − ˜ M / h c − vol ∧ d ˜ h − + c − / d (cid:0) e − − Φ ∞ ) ˜ φ (cid:1)i . (4.21)Here tildes on ˜ ∗ and ˜ φ indicate that the expressions are computed with the metric d ˜ s . We can now show that in this solution the limit c → ∞ gives a solution of the In the expressions below Φ ∞ enters only in the combination Φ − Φ ∞ , which is finite in the limit. c the metric for each X ij solution reads d ˆ s str = ˜ M h ˜ h − / c − dx + ˜ h / c (cid:0) ds ( X i ) + O ( c − ) (cid:1)i . (4.22)From the differential equation for the dilaton we find thatΦ ′ = c − (cid:0) H ′ + O (cid:0) c − (cid:1)(cid:1) (4.23)where H is the warp factor of the solution found in [31], which reads H = 3( r + r )4 r r ( r + r r + r ) (cid:0) r + 24 r r + 48 r r + 47 r r + 54 r r + 36 r r + 18 r r + 9 r (cid:1) + 8 √ r arctan 2 r + r √ r + q . (4.24) q is an integration constant and taking q = − √ πr we have that H ∼ / (4 r ) when r → ∞ . Solving for the dilaton in an expansion in c − we find e − Φ ∞ ) = 1 + c − H + O (cid:0) c − (cid:1) . (4.25)Notice that although this was obtained in [31] for the G holonomy metric on X , itfollows from our discussion in Section 3.4, that this expression is invariant under Σ and hence the same function H in (4.24) appears for any X i . Thus taking the limit c → ∞ on a X ij solution, gives the following solution d ˆ s str = ˜ M (cid:2) H − / dx + H / ds ( X i ) (cid:3) , e = H / , ˆ F = − ˜ M / h vol ∧ d ˜ H − − ∗ L i , ˆ H = L , (4.26)where L is a harmonic three-form on X i . This is precisely the warped G solutionpresented in [31]. Notice that asymptotically the string frame metric goes to AdS × Y ,where Y ∼ = S × S , however the dilaton vanishes like e ∼ / (2 r ). In fact, by setting r = 0 we have the exact solution with metric d ˆ s str = 92 ˜ M (cid:2) ds (AdS ) + ds ( Y ) (cid:3) , (4.27) e = 9 / (2 r ) and non-trivial F and H fluxes. However, the solution does not haveconformal symmetry because the dilaton depends on the radial coordinate r . In (4.27) This can be extracted from the c → ∞ limit of c − / ˜ ∗ d ˜ φ .
31e can replace Y ∼ = S × S with another nearly K¨ahler metric, provided there existsthe appropriate harmonic three-form L on the G cone, thus obtaining a solutiongenerically preserving the same amount of supersymmetry. These metrics are in factsolutions of massive Type IIA supergravity [38]. This is a curious fact that might berelevant for AdS/CFT applications [39].In conclusion, for any solution of the type of [31], arising from configurations of D2branes and fractional NS5 branes transverse to a G manifold X i , we have constructed aone-parameter family of deformations, with the same AdS × Y asymptotics. These areanalogous to the baryonic branch deformations [11] of the Klebanov-Strassler solution[8]. In particular, they break the Z ⊂ Σ symmetry of a G holonomy manifold X i . In this paper we have discussed various supergravity solutions related to configurationsof fivebranes wrapping a three-sphere in a G holonomy manifold X i ∼ = S × R . Ourbasic solutions are examples of torsional G manifolds [27] and comprise some casespreviously studied in [33, 23, 22]. There are six solutions characterised by a non-trivialparameter. As we change this parameter, each solution interpolates between a G manifold with (NS5 or D5) branes on a three-sphere and a distinct G manifold with(NS or RR) flux on a different three-sphere. This is then an explicit realisation ofa geometric transition between a pair of G manifolds, analogous to the version ofthe conifold transition described in [14]. The six solutions pairwise connect the threebranches of the classical moduli space of G holonomy metrics on S × R [25]. It wouldbe interesting to see if the picture that we discussed, which is purely classical, may berelated to a “large N duality” similar to [16].From each of the basic solutions we constructed new Type IIA backgrounds withD2 brane charge by employing a simple generating method applicable to a class ofgeometries with interpolating G structure. The solutions constructed in this way areone-parameter deformations of the solutions presented in [31], corresponding to D2branes and fractional NS5 branes transverse to the G manifold S × R . Therefore,they are analogous to the baryonic branch deformation [11] of the Klebanov-Strasslersolution [8].Based purely on supergravity considerations, it is natural to expect a close relationbetween the N = 1 Chern-Simons theory discussed in [23] and the N = 1 three-dimensional field theory dual to the solutions above. Let us conclude with some specu-32ations about the field theory duals. First of all, the existence of a finite size S in thegeometry suggests that the IR field theory should be confining, as in [23]. A standardcomputation of the number of D2 branes N shows that this is running, and vanishes inthe IR, as the number of D3 branes in [8]. This suggests that the three-dimensionalfield theory may be a quiver with gauge group U ( N ) × U ( N + M ). Moreover, we expectthat the three-form flux H will induce Chern-Simons terms, like in [23]. We also havea running C field on a three-sphere at infinity, analogous to the B field in [8], with k = R C ∼ M t , suggesting the relation N = kM . A possible scenario is thereforethat the solution of [31] describes a “cascading” three-dimensional quiver, which in the“last step” becomes the U ( M ) M/ theory of [23]. However, an important caveat is thatour solutions are related to NS5 branes in Type IIA, while the discussion in [23] ap-plies to configurations of Type IIB NS5 branes. Nevertheless, the relation between thevarious solutions based on the conifold and the solutions we discussed here indicatesthat a precise connection between the four-dimensional “parent” field theories and thethree-dimensional field theories should exist, along the lines of [40, 41]. In particular,we can uplift our solutions to M-theory and subsequently reduce along a U (1) insidethe non-trivial geometry, thus obtaining solutions with the topology of the deformedor resolved conifold [26] (times S ).Another relationship between the U ( M ) M/ Chern-Simons theory and the putativefield theory dual to [31] is suggested by the one-parameter deformations of [31] that wedescribed. In particular, we saw how in a certain regime of the parameter ( c ∼ / C fieldon the three-sphere wrapped by these branes suggests that perhaps the relevant fieldtheory is the theory on fivebranes wrapped on a fuzzy three-sphere [14].We leave the investigation of these ideas for future work. Acknowledgements
We are very grateful to Juan Maldacena, Carlos N´u˜nez, Johannes Schmude and JamesSparks for discussions and useful comments. We also thank Diego Rodriguez-Gomezfor comments and collaboration on related topics. D. M. is partially supported by anEPSRC Advanced Fellowship EP/D07150X/3. Here the running is not logarithmic, but we have N ( t ) ∼ M t at large t . SU (2) invariant one-forms Consider three elements a , a , a ∈ SU (2) obeying the constraint a a a = 1. Wedefine the following SU (2) Lie-algebra valued one-forms a − da ≡ i α i τ i a da − ≡ i β i τ i a − da ≡ − i γ i τ i (A.1)where τ i are Pauli matrices. We can invert these obtaining α i = − i Tr[ τ i a − da ] β i = − i Tr[ τ i a da − ] γ i = i Tr[ τ i a − da ] (A.2)Parameterising the group elements explicitly in terms of angular variables as a = e − iφ τ / e − iθ τ / e − iψ τ / a = e iψ τ / e iθ τ / e iφ τ / a = a − a − = e − iφ τ / e − iθ τ / e − i ( ψ − ψ ) τ / e iθ τ / e iφ τ / (A.3)after some computation we get α + iα = − e − iψ ( dθ + i sin θ dφ ) , α = − ( dψ + cos θ dφ ) ,β + iβ = − e − iψ ( dθ + i sin θ dφ ) , β = − ( dψ + cos θ dφ ) . (A.4)Notice α i and β i are SU (2) left-invariant one-forms, obeying dα = + α ∧ α , dβ = + β ∧ β , (A.5)and cyclic permutations. We can also define the following Lie-algebra valued one-forms a da − ≡ i α i τ i a − da ≡ i β i τ i (A.6)A similar computation gives˜ α + i ˜ α = e iφ ( dθ − i sin θ dψ ) , ˜ α = dφ + cos θ dψ , ˜ β + i ˜ β = e iφ ( dθ − i sin θ dψ ) , ˜ β = dφ + cos θ dψ . (A.7)34hese are SU (2) right-invariant one-forms, obeying d ˜ α = + ˜ α ∧ ˜ α , d ˜ β = + ˜ β ∧ ˜ β , (A.8)and cyclic permutations. Computing the γ i we obtain − γ i = ˜ α i + M ij β j (A.9)where M ij is the following SO (3) matrix M ij = cos φ cos ψ − cos θ sin φ sin ψ − cos θ cos ψ sin φ − cos φ sin ψ sin θ sin φ cos ψ sin φ + cos θ cos φ sin ψ cos θ cos φ cos ψ − sin φ sin ψ − cos φ sin θ sin θ sin ψ cos ψ sin θ cos θ (A.10) We note the following identities X i α i = X i ˜ α i , X i β i = X i ˜ β i , (A.11)and X i γ i = X i ( α i − β i ) . (A.12)To prove the latter we have to use M ij M ik = δ ik and α i = − M ji ˜ α j . We identify theabove with the (left-invariant) one-forms σ i and Σ i used in the main text σ i = − α i = i Tr[ τ i a − da ]Σ i = − β i = i Tr[ τ i a da − ] (A.13)where the minus signs have been included in order to match with our conventions onthe Lie-algebra relations dσ = − σ ∧ σ , etcetera. Notice that γ i = i Tr[ τ i a − da ] = ˜ σ i + M ij Σ j = M ij (Σ j − σ j ) . (A.14)We also define da = − X i (Tr[ τ i a − da ]) da = − X i (Tr[ τ i a da − ]) da = − X i (Tr[ τ i a − da ]) (A.15)35 Derivation of the BPS equations
In the following we derive the BPS system (3.7) from the G structure equations (3.2).Recall the metric ansatz is ds = M (cid:2) dt + a X i =1 σ i + b X i =1 (Σ i − (1 + ω ) σ i ) (cid:3) . (B.1)In this section we define the orthonormal frame with an extra factor of √ M withrespect to the definition (2.14), namely here e t = √ M dt ˜ e a = √ M a σ a e a = √ M b (Σ a − (1 + ω ) σ a ) (B.2)so that the associative three form in the following is defined as φ = e t ∧ J + Re[ e iθ Ω] (B.3)and we use the definitions in (2.16). Let us look at the first equation in (3.2). We firstcompute some useful intermediate results: √ M dJ = ddt log( ab ) e t ∧ J + 3 b a (1 − ω ) 13! ǫ abc ˜ e a ˜ e b ˜ e c − ωa ǫ abc ˜ e a ˜ e b e c − b ǫ abc e a e b ˜ e c (B.4) √ M d ( J ∧ J ) = ddt log( ab ) e t ∧ J ∧ J (B.5) √ M d
Re[Ω] = ddt log( b ) e t ǫ abc e a e b e c + 3 bω ′ a e t ǫ abc ˜ e a ˜ e b ˜ e c − ddt log( a b ) e t ǫ abc ˜ e a ˜ e b e c − bω ′ a e t ǫ abc e a e b ˜ e c − (cid:18) b + b a (1 − ω ) (cid:19) J ∧ J (B.6) √ M d
Im[Ω] = − ddt log a e t ǫ abc ˜ e a ˜ e b ˜ e c + ddt log( ab ) e t ǫ abc e a e b ˜ e c − bω ′ a e t ǫ abc e a ˜ e b ˜ e c − ω a J ∧ J (B.7)where we used the identity e e ˜ e ˜ e + cyclic = − J ∧ J . (B.8)36fter some more algebra we find √ M φ ∧ dφ = e t ∧ J ∧ J ∧ J (cid:20) b a ω ′ + 23 θ ′ + sin θ ωa − cos θ (cid:18) b + b a (1 − ω ) (cid:19) (cid:21) . (B.9)Thus the first equation in (3.2) implies b a ω ′ + 23 θ ′ + sin θ ωa − cos θ (cid:18) b + b a (1 − ω ) (cid:19) = 0 (B.10)Let us now look at the second equation in (3.2). We first calculate √ M d ∗ φ = e t ∧ J ∧ J (cid:20) ddt log( ab ) − cos θ ωa − sin θ (cid:18) b + b a (1 − ω ) (cid:19) (cid:21) . (B.11)The equation d ( e − ∗ φ ) = 0 may be written as d ∗ φ = 2 d Φ ∧ ∗ φ (B.12)and after writing d Φ = M − / Φ ′ e t , it may be regarded as giving the derivative of thedilaton in terms of the remaining functions. In particular2Φ ′ = ddt log a b − cos θ ωa − sin θ (cid:18) b + b a (1 − ω ) (cid:19) . (B.13)Finally we analyse the last equation in (3.2). This can be rewritten as ∗ H = 2 d Φ ∧ φ − dφ = 2 M − / Φ ′ e t ∧ (cos θ Re[Ω] − sin θ Im[Ω]) − dφ (B.14)Then we compute dφ : √ M dφ = e t ǫ abc e a e b e c (cid:20) ddt (cos θ ) + cos θ ddt log b (cid:21) ++ e t ǫ abc ˜ e a ˜ e b ˜ e c (cid:20) ddt (sin θ ) − b a (1 − ω ) + 3 cos θ b a ω ′ + sin θ ddt log a (cid:21) + e t ǫ abc ˜ e a ˜ e b e c (cid:20) − ddt (cos θ ) + ωa + 2 sin θ b a ω ′ − cos θ ddt log( a b ) (cid:21) + e t ǫ abc e a e b ˜ e c (cid:20) − ddt (sin θ ) + 1 b − cos θ b a ω ′ − sin θ ddt log( ab ) (cid:21) + 12 J ∧ J (cid:20) sin θ ωa − cos θ (cid:16) b + b a (1 − ω ) (cid:17)(cid:21) . (B.15)37e now need to compute ∗ H . First, starting from the ansatz (3.4) and using therelations (3.17) we obtain:4 √ M H = − ǫ abc e a e b e c k b + 13! ǫ abc ˜ e a ˜ e b ˜ e c a (cid:20) k − k ω (3 + ω ) − γ (1 − ω ) (cid:21) + 12! ǫ abc ˜ e a ˜ e b e c a b (cid:20) − k (1 + ω ) + 2 ωγ (cid:21) − ǫ abc e a e b ˜ e c ab ( k ω − γ ) − e t ∧ J γ ′ ab (B.16)Then the Hodge dual is:4 √ M ∗ H = − e t ǫ abc e a e b e c a h k − k ω (3 + ω ) − γ (1 − ω ) i − e t ǫ abc ˜ e a ˜ e b ˜ e c k b + e t ǫ abc ˜ e a ˜ e b e c ab ( k ω − γ )+ e t ǫ abc e a e b ˜ e c a b h − k (1 + ω ) + 2 ωγ i − J ∧ J γ ′ ab . (B.17)38utting everything together we find: a ′ = 16 ab sin θ (cid:0) − ω (cid:1) + 4 a cos θ (cid:0) γ − k ω (cid:1) + 4 ab sin θ (cid:0) k + k ω − ωγ (cid:1) a b − b cos θ (cid:0) k − k ω − a ω − k ω − γ + 3 ω γ (cid:1) a b b ′ = 32 ab sin θ (cid:0) ω − (cid:1) + 4 a (cid:0) cos θ − θ (cid:1)(cid:0) k ω − γ (cid:1) a b − ab sin θ (cid:0) a + k + k ω − ωγ (cid:1) a b + b sin θ tan θ (cid:0) − k + 3 k ω + 96 a ω + k ω + 3 γ − ω γ (cid:1) a bω ′ = 32 a b cos θ + 4 a sin θ (cid:0) k ω − γ − b ω (cid:1) a b + b sin θ (cid:0) k − k ω − k ω − γ + 3 ω γ (cid:1) a b + 4 ab cos θ (cid:0) − b ( ω −
1) + k + k ω − ωγ (cid:1) a b γ ′ = − a cos θ + 8 ab sin θω + 2 b cos θ (cid:0) ω − (cid:1) a (B.18)where the angle θ is fixed in terms of the other functions and reads:cot θ = b (cid:16) a (cid:0) γ − k ω (cid:1) + b (cid:0) − k + 3 k ω + 96 a ω + k ω + 3 γ − ω γ (cid:1)(cid:17) a (cid:16) − a (cid:0) k − b (cid:1) + 3 b (cid:0) b (1 − ω ) + k + k ω − ωγ (cid:1)(cid:17) (B.19)Finally, substituting the functions a , b and ω with the functions f i , and using a com-puter program to simplify the expressions, we find the BPS system (3.7). C Supersymmetry conditions in Type IIA
C.1 Reduction from d = 11 General conditions characterising N = 1 solutions of eleven-dimensional supergravityof the warped product type X × w M where X is either R , or AdS , werepresented in [35]. Here we are interested in the case that X = R , . The eleven-dimensional metric is written as d ˆ s = e ( dx + ds ) (C.1)39nd the four-form flux reads G = e ( F + vol ∧ f ) . (C.2)Thus F is a four-form and f is a one-form. Upon setting m = 0, the equations (3.11)- (3.16) of [35] become d ( e K cos ζ ) = 0 K ∧ d ( e ∗ φ ) = 0 d ( e vol cos ζ ) = 0 dφ ∧ φ cos ζ = 2 ∗ (cos ζ f − dζ ) (C.3)Here φ is a three-form, K is a one-form and ζ is a function, defined as spinor bilinears,that characterise the G structure in eight dimensions. The seven-dimensional Hodgestar operator is defined as ∗ = i K ∗ and vol = φ ∧ ∗ φ . The electric and magneticfluxes are the determined in terms of the G structure as e − d ( e sin ζ ) = fe − d ( e cos ζ φ ) = − ∗ F + sin ζ F (C.4)The latter equation obeyed by the magnetic flux F may be inverted givingcos ζ F = − e − (cid:2) sin ζ d ( e cos ζ φ ) + ∗ d ( e cos ζ φ ) (cid:3) . (C.5)The one-form K in general does not correspond to a Killing vector. However, in orderto reduce to Type IIA, we will assume that the dual vector K is Killing. In particular,writing K = e / − ∆ dy , the eleven-dimensional metric takes the form d ˆ s = e ( dx + ds + e / − dy ) (C.6)and its reduction to ten dimensions then can be simply read off: ds str = e / ( dx + ds ) . (C.7)Then we write d = d + dy∂ y , f = f + dyf y . (C.8)Looking first at the fluxes we find f y = 0 ,f = e − d ( e sin ζ ) . (C.9)40sing these equations, from (C.3) we obtain d (cid:0) e ∗ φ (cid:1) = 0 φ ∧ d φ = 02 d ζ − e − cos ζ d (cid:0) e sin ζ (cid:1) = 0 d (cid:0) e / cos ζ (cid:1) = 0 (C.10)Finally, the reduction of the four-form G gives the NS three-form H and the RRfour-form F : H = 1cos ζ e − / ∗ d (cid:0) e cos ζ φ (cid:1) ,F = vol ∧ d (cid:0) e sin ζ (cid:1) − sin ζ cos ζ e − d (cid:0) e cos ζ φ (cid:1) . (C.11) C.2 Killing spinor ansatze
In this Appendix we discuss ansatze for the Killing spinors in eleven and ten dimensions.Although in the main text we have not derived the equations from the Killing spinorsdirectly, it may be useful to spell out some details about spinors and representationsof gamma matrices. The general spinor ansatz in eleven dimensions reads [35] η = e ∆ / ψ ⊗ ξ = e ∆ / ψ ⊗ ( ξ + + ξ − ) . (C.12)We use the following explicit representation of gamma matricesˆΓ µ = − e − ∆ ρ µ ⊗ ˆ γ χ µ = 0 , . . . , m = e − ∆ ⊗ ˆ γ m m = 3 , . . . ,
10 (C.13)The ˆ γ m are 16 ×
16 gamma-matrices and ˆ γ χ is the chirality matrix in d = 8, with (ˆ γ χ ) = . The ρ µ denote 2 × d = 1 + 2. In an explicit representationthese may be taken [35] as follows ρ = iσ , ρ = σ , ρ = σ , (C.14)where σ i are Pauli matrices. The Majorana condition in eleven dimensions η = η c = D η ∗ , with D = σ ⊗ implies that ψ ∗ = σ ψ and ξ ± = ξ ∗± . Hence ψ is a Majoranaspinor in d = 1 + 2 and ξ ± are Majorana-Weyl spinors in d = 8.To make the reduction to ten dimensions it is convenient to chose the followingrepresentation of d = 8 gamma-matrices:ˆ γ m = ( ˆ γ i = σ ⊗ γ i i = 3 , . . . , γ = σ ⊗ (C.15)41hese are real and symmetric, taking γ i to be purely imaginary and anti-symmetric.Then the d = 8 chirality matrix isˆ γ χ = ˆ γ · · · ˆ γ = − σ ⊗ (C.16)and is again real and symmetric. We then write the d = 8 Majorana-Weyl spinors as ξ + = α + ⊗ β + , ξ − = α − ⊗ β − , (C.17)where α ± are two-component spinors and β ± are eight-component spinors, which canall be taken to be real. Imposing the d = 8 chirality conditions ˆ γ χ ξ ± = ± ξ ± thenimplies σ α ± = ∓ α ± . (C.18)Thus, up to an overall real function, we can take α + = (cid:18) − (cid:19) , α − = (cid:18) (cid:19) . (C.19)Upon reducing to Type IIA, the complexified Killing spinor is simply related to theKilling spinor in eleven dimension as η = e − Φ / ǫ , hence the ansatz may be written as ǫ = ǫ + ǫ = e Φ / / ψ ⊗ ( θ ⊗ χ + θ ⊗ χ ) . (C.20)The ten-dimensional gamma matrices are the same as the eleven-dimensional ones upto a warp factor, where the eleventh one becomes the ten-dimensional chirality matrix.Namely Γ µ = e − ∆ − Φ / ρ µ ⊗ σ ⊗ µ = 0 , . . . , i = e − ∆ − Φ / ⊗ σ ⊗ γ i i = 3 , . . . , χ = ⊗ σ ⊗ (C.22)where we have used the convention that γ γ · · · γ = + i . (C.23)The Majorana condition on the complexified ten-dimensional spinor is ǫ = ˜ D ǫ ∗ , wherewe can take ˜ D = σ ⊗ ⊗ . Therefore we have ψ ∗ ⊗ θ ∗ ⊗ χ ∗ = σ ψ ⊗ θ ⊗ χ ψ ∗ ⊗ θ ∗ ⊗ χ ∗ = σ ψ ⊗ θ ⊗ χ (C.24) Taking γ γ · · · γ = − i gives Γ χ = − ⊗ σ ⊗ . σ ψ = ψ ∗ , θ i = θ ∗ i , χ i = χ ∗ i . (C.25)The ten-dimensional chirality conditions giveΓ χ ǫ = ǫ ⇒ σ θ = θ Γ χ ǫ = − ǫ ⇒ σ θ = − θ (C.26)Therefore, up to overall factors (not necessarily constant) that we can reabsorb in χ i ,we can take θ = (cid:18) (cid:19) , θ = (cid:18) (cid:19) . (C.27)Comparing (C.12) with (C.20) we find ξ + = 12 (cid:18) − (cid:19) ⊗ ( χ − χ ) ξ − = 12 (cid:18) (cid:19) ⊗ ( χ + χ ) (C.28)Now using ξ T + ξ + + ξ T − ξ − = 2 ξ T + ξ + − ξ T − ξ − = 2 sin ζ (C.29)from [35], we compute χ T χ + χ T χ = 2 χ T χ = − sin ζ (C.30)Since we are interested in a strict G structure, we must have χ ∝ χ . In fact, if χ and χ were not parallel, we could construct a vector χ T γ i χ , reducing the structureto SU (3). Then we have that χ = √ α χχ = √ α χ where we normalised the spinor χ as χ T χ = 1 and sin 2 α = − sin ζ . 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