AdS_5 Solutions of Type IIB Supergravity and Generalized Complex Geometry
Maxime Gabella, Jerome P. Gauntlett, Eran Palti, James Sparks, Daniel Waldram
aa r X i v : . [ h e p - t h ] F e b Imperial-TP-2009-JG-05October 29, 2018
AdS Solutions of Type IIB Supergravityand Generalized Complex Geometry
Maxime Gabella , Jerome P. Gauntlett , Eran Palti ,James Sparks and Daniel Waldram Rudolf Peierls Centre for Theoretical Physics,University of Oxford,1 Keble Road, Oxford OX1 3NP, U.K. Theoretical Physics Group, Blackett Laboratory,Imperial College, London SW7 2AZ, U.K.The Institute for Mathematical Sciences,Imperial College, London SW7 2PE, U.K. Mathematical Institute, University of Oxford,24-29 St Giles’, Oxford OX1 3LB, U.K.
Abstract
We use the formalism of generalized geometry to study the generic super-symmetric
AdS solutions of type IIB supergravity that are dual to N = 1superconformal field theories (SCFTs) in d = 4. Such solutions have an asso-ciated six-dimensional generalized complex cone geometry that is an extensionof Calabi-Yau cone geometry. We identify generalized vector fields dual to thedilatation and R -symmetry of the dual SCFT and show that they are general-ized holomorphic on the cone. We carry out a generalized reduction of the coneto a transverse four-dimensional space and show that this is also a generalizedcomplex geometry, which is an extension of K¨ahler-Einstein geometry. Remark-ably, provided the five-form flux is non-vanishing, the cone is symplectic. Thesymplectic structure can be used to obtain Duistermaat-Heckman type integralsfor the central charge of the dual SCFT and the conformal dimensions of op-erators dual to BPS wrapped D3-branes. We illustrate these results using thePilch-Warner solution. Introduction
Supersymmetric
AdS × Y solutions of type IIB supergravity, where Y is a compactRiemannian manifold, are dual to supersymmetric conformal field theories (SCFTs)in d = 4 spacetime dimensions with (at least) N = 1 supersymmetry. An importantspecial subclass is when Y is a five-dimensional Sasaki-Einstein manifold SE , andthe only non-trivial flux is the self-dual five-form. Recall that, by definition, the six-dimensional cone metric with base given by the SE space is a Calabi-Yau cone, andthat the dual SCFT arises from D3-branes located at the apex of this cone. There hasbeen much progress in understanding the AdS/CFT correspondence in this setting. Forexample, there are rich sets of explicit SE metrics [1]-[3], and there are also powerfulconstructions using toric geometry. Moreover, for the toric case, the correspondingdual SCFTs have been identified, e.g. [4]-[7].A key aspect of this progress has been the appreciation that the abelian R -symmetry,which all N = 1 SCFTs in d = 4 possess, contains important information aboutthe SCFT. For example, the a central charge is fixed by the R -symmetry, as are theanomalous dimensions of (anti-)chiral primary operators [8]. It is also known that the R -symmetry can be identified via the procedure of a -maximization, which, roughly, saysthat the correct R -symmetry is the one that maximizes the value of a over all possibleadmissible R -symmetries [9]. For the solutions of type IIB supergravity with Y = SE ,the R -symmetry manifests itself as a canonical Killing vector ξ on SE . This defines aKilling vector on the Calabi-Yau cone, also denoted by ξ , which is a real holomorphicvector field. The Calabi-Yau cone is K¨ahler, and hence symplectic, and Y admits acorresponding contact structure for which ξ is the Reeb vector. When Y = SE the a central charge is inversely proportional to the volume of SE , and in [10, 11] severalgeometric formulae for a in terms of ξ were derived. Analogous geometric formulae forthe conformal dimension of the chiral primary operator dual to a D3-brane wrappedon a supersymmetric submanifold Σ ⊂ Y were also presented. Of particular interesthere are the formulae that show that using symplectic geometry these quantities can bewritten as Duistermaat-Heckman integrals on the cone and hence can be evaluated bylocalization. In addition to providing a geometrical interpretation of a-maximization,these formulae and others in [10, 11] also provide practical methods for calculatingquantities of physical interest without needing the full explicit Sasaki-Einstein metric(which, apart from some special classes of solution, remains out of reach).The focus of this paper is on AdS × Y solutions with Y more general than SE .1ost known solutions are actually part of continuous families of solutions containinga Sasaki-Einstein solution and correspond to exactly marginal deformations of thecorresponding SCFT. For example, starting with a toric SE solution one can constructnew β -deformed solutions using the techniques of [12]. There is also the “Pilch-Warnersolution” explicitly constructed in [13] (based on [14]). It has been shown numericallyin [15] that the Z orbifold of the Pilch-Warner solution is part of a continuous familyof solutions that includes the Sasaki-Einstein AdS × T , solution. Using the resultsof [16, 17] this should be part of a larger family of continuous solutions that are yetto be found. Similarly, in addition to the β -deformations of the AdS × S solutionthere are additional deformations [16] that are also not yet constructed (a perturbativeanalysis was studied in [18]). Having a better understanding of the geometry underlyinggeneral AdS × Y solutions could be useful for finding these deformed solutions butmore generally could be useful in constructing new solutions that are not connectedwith Sasaki-Einstein geometry at all.The first detailed analysis of supersymmetric AdS × Y solutions of type IIB super-gravity, for general Y with all fluxes activated, was carried out in [19]. The conditionsfor supersymmetry boil down to a set of Killing spinor equations on Y for two spinors(when Y = SE there is only one such spinor). By analysing these equations a setof necessary and sufficient conditions for supersymmetry were established. In light ofthe progress summarized above for the Sasaki-Einstein case, it is natural to investigatethe associated geometry of the cone over Y , and that is the principal purpose of thispaper.As we shall discuss in detail, the cone X admits a specific kind of generalized complexgeometry. Aspects of this geometry, restricting to a class of SU (2)-structures, were firststudied in [20, 21]. By viewing AdS × Y as a supersymmetric warped product R , × X ,one sees immediately that the cone admits two compatible generalized almost complexstructures [22, 23], or equivalently two compatible pure spinors, Ω ± . In fact dΩ − = 0,so that Ω − defines an integrable generalized complex geometry, while dΩ + is relatedto the RR flux. The cone is thus generalized Hermitian, and it is also generalizedCalabi-Yau in the sense of [24].Here, we will identify a generalized vector field ξ on the cone that is dual to the R -symmetry and another that is dual to the dilatation symmetry of the dual SCFTand show that they are both generalized holomorphic vector fields on the cone (withrespect to the integrable generalized complex structure). This precisely generalizesknown results for the Sasaki-Einstein case. We also note that all supersymmetric2 dS × Y solutions satisfy the condition of [20] that there is an SU (2)-structure onthe cone.In the Sasaki-Einstein case, one can carry out a symplectic reduction of the Calabi-Yau cone to obtain a four-dimensional transverse K¨ahler-Einstein space which, in gen-eral, is only locally defined. Constructing locally defined K¨ahler-Einstein spaces hasbeen a profitable way to construct Sasaki-Einstein manifolds, e.g. [25]. Here we willshow, using the formalism of [26, 27], that for general Y there is an analogous reduc-tion of the corresponding six-dimensional generalized Calabi-Yau cone geometry to afour-dimensional space, which again is only locally defined in general, that is general-ized Hermitian. More precisely, the four-dimensional geometry admits two compatiblegeneralized almost complex structures, one of which is integrable.We present explicit expressions for the pure spinors Ω ± associated with the six-dimensional cone in terms of the Killing spinor bilinears presented in [19]. We shallcomment upon how Ω − , associated with the integrable generalized complex structure,contains information on the mesonic moduli space of the dual SCFT and also, briefly, onsome relations connected with generalized holomorphic objects and dual BPS operators.By analysing the pure spinor Ω + , associated with the non-integrable complex struc-ture, and focusing on the case when the five-form flux is non-vanishing, we show that,perhaps somewhat surprisingly, the cone is symplectic. We shall see that Y is a contactmanifold and that the vector part, ξ v , of the generalized vector ξ , which also definesa Killing vector on Y , is the Reeb vector field associated with the contact structure.We show that the symplectic structure can be used to obtain Duistermaat-Heckmantype integrals for the central charge a of the dual SCFT and also for the conformaldimensions of operators dual to wrapped BPS D3-branes. Once again these resultsprecisely generalize those for the Sasaki-Einstein case. Some of these results were firstpresented in [28]; here we will provide additional details and also show how they arerelated to the generalized geometry on the cone.Finally, we will illustrate some of our results using the Pilch-Warner solution. Thepaper begins with a review of generalized geometry and it ends with three appendicescontaining some details about our conventions, some technical derivations, and a briefdiscussion of the Sasaki-Einstein case. 3 Generalized geometry
We begin by reviewing some aspects of generalized complex geometry [24], to fix con-ventions and notation. For further details see, for example, [29].
Generalized geometry starts with the generalized tangent bundle E over a manifold X ,which is a particular extension of T X by T ∗ X obtained by twisting with a gerbe . Agerbe is simply a higher degree version of a U (1) bundle with unitary connection. Justas topologically a U (1) bundle is determined by its first Chern class, the topology ofa gerbe is determined by a class in H ( X, Z ). To define a gerbe [30], one begins withan open cover { U i } of X together with a set of functions g ijk : U i ∩ U j ∩ U j → U (1)defined on triple overlaps. These are required to satisfy g ijk = g jik − = g ikj − = g kji − ,together with the cocycle condition g jkl g ikl − g ijl g ijk − = 1 on quadruple overlaps. A connective structure [30] on a gerbe is a collection of one-forms Λ ( ij ) defined on doubleoverlaps U i ∩ U j satisfying Λ ( ij ) + Λ ( jk ) + Λ ( ki ) = − (2 π i l s ) g − ijk d g ijk on triple overlaps.In turn, a curving is a collection of two-forms B ( i ) on U i satisfying B ( j ) − B ( i ) = dΛ ( ij ) . (2.1)It follows that d B ( j ) = d B ( i ) = H is a closed global three-form on X , called the curvature , and, in cohomology, πl s ) H ∈ H ( X, Z ) (in the normalization that we shalluse in this paper). In string theory, the collection of two-forms B ( i ) , which we writesimply as B , is the NS B -field and H is its curvature.The generalized tangent bundle E is an extension of T X by T ∗ X −→ T ∗ X −→ E π −→ T X −→ . (2.2)Locally, sections of E , which we refer to as generalized tangent vectors, may be writtenas V = x + λ , where x ∈ Γ( T X ) and λ ∈ Γ( T ∗ X ). More precisely, in going from onecoordinate patch U i to another U j the extension is defined by the connective structure x ( i ) + λ ( i ) = x ( j ) + (cid:0) λ ( j ) − i x ( j ) dΛ ( ij ) (cid:1) . (2.3)The bundle E is in fact isomorphic to T X ⊕ T ∗ X . However, the isomorphism isnot canonical but depends on a choice of splitting, defined by a two-form curving B satisfying (2.1). It follows that x + ( λ − i x B ) ∈ Γ( T X ⊕ T ∗ X ) . (2.4)4hus the definition (2.3) of E can be viewed as encoding the patching of a class oftwo-form curvings B .Writing d = dim R X , there is a natural O ( d, d )-invariant metric h· , ·i on E , given by h V, W i = ( i x µ + i y λ ) , (2.5)where V = x + λ , W = y + µ , or in two-component notation, h V, W i = (cid:0) x λ (cid:1) ! yµ ! . (2.6)This metric is invariant under O ( d, d ) transformations acting on the fibres of E , defininga canonical O ( d, d )-structure. A general element O ∈ O ( d, d ) may be written in termsof d × d matrices a , b , c , and d as O = a bc d ! , (2.7)under which a general element V ∈ E transforms by V = xλ ! OV = a bc d ! xλ ! . (2.8)The requirement that h OV , OV i = h V, V i implies a T c + c T a = 0, b T d + d T b = 0 and a T d + c T b = 1. Note that the GL ( d ) action on the fibres of T X and T ∗ X embeds as asubgroup of O ( d, d ). Concretely it maps V V ′ = a a − T ! xλ ! , (2.9)where a ∈ GL ( d ). Given a two-form ω , one also has the abelian subgroupe ω = ω ! such that V = x + λ V ′ = x + ( λ − i x ω ) . (2.10)This is usually referred to as a B -transform. Given a bivector β one can similarlydefine another abelian subgroup of β -transformse β = β ! such that V = x + λ V ′ = ( x + i λ β ) + λ . (2.11)5ote that the patching (2.3) corresponds to a B -transform with ω = dΛ ( ij ) . Similarly,the splitting isomorphism between E and T X ⊕ T ∗ X defined by B is also a B -transform E e B ⇄ e − B T X ⊕ T ∗ X . (2.12)There is a natural bracket on generalized vectors known as the Courant bracket,which encodes the differentiable structure of E . It is defined as[ V, W ] = [ x + λ, y + µ ] = [ x, y ] Lie + L x µ − L y λ − d ( i x µ − i y λ ) , (2.13)where [ x, y ] Lie is the usual Lie bracket between vectors and L x is the Lie derivativealong x . The Courant bracket is invariant under the action of diffeomorphisms and B -shifts ω that are closed, d ω = 0, giving an automorphism group which is a semi-directproduct Diff( X ) ⋉ Ω ( X ). Note, however, that in string theory only B -shifts bythe curvature of a unitary line bundle on X are gauge symmetries, as opposed to shiftsby arbitrary closed two-forms, leading to a smaller automorphism group. Under aninfinitesimal diffeomorphism generated by a vector field x and a B -shift with ω = d λ ,one has the generalized Lie derivative by V = x + λ on a generalized vector field W = y + µ δW ≡ L V W = [ x, y ] Lie + ( L x µ − i y d λ ) . (2.14)This is also known as the Dorfman bracket [ V, W ] D , the anti-symmetrization of whichgives the Courant bracket (2.13). Note that since the metric h· , ·i is invariant under O ( d, d ) transformations its generalized Lie derivative vanishes. Given a particularchoice of splitting (2.1) defined by B , the Courant bracket on E defines a Courantbracket on T X ⊕ T ∗ X , known as the twisted Courant bracket. It is given by[ x + λ, y + µ ] H = e B [e − B ( x + λ ) , e − B ( y + µ )]= [ x + λ, y + µ ] + i y i x H , (2.15)where by an abuse of notation we are writing x + λ and y + µ for sections of T X ⊕ T ∗ X whereas above they were sections of E .Given the metric h· , ·i , one can define Spin ( d, d ) spinors in the usual way. Since thevolume element in Cliff( d, d ) squares to one, one can define two helicity spin bundles S ± ( E ) as the ± S ± ( E ) on U i can be identified with a even- or odd-degree polyform Ω ± ∈ Ω even/odd ( X )restricted to U i , with the Clifford action of V ∈ Γ( E ) given by V · Ω ± = i x Ω ± + λ ∧ Ω ± . (2.16)6t is easy to see that ( V · W + W · V ) · Ω ± = 2 h V, W i Ω ± , (2.17)as required. Using this Clifford action the B -transform (2.10) on spinors is given byΩ ± e ω Ω ± , (2.18)where the exponentiated action is by wedge product. The patching (2.3) of E thenimplies that Ω ( i ) ± = e dΛ ( ij ) Ω ( j ) ± . (2.19)Furthermore a splitting B also induces an isomorphism between S ± ( E ) and S ± ( T X ⊕ T X ∗ ) S ± ( E ) e B ⇄ e − B S ± ( T X ⊕ T ∗ X ) , (2.20)again by the action of the exponentiated wedge product. If Ω ± is a section of S ± ( E ),we will sometimes write Ω B ± ≡ e B Ω ± for the corresponding section of S ± ( T X ⊕ T ∗ X )defined by the splitting B . The real Spin ( d, d )-invariant spinor bilinear on sections of S ± ( E ) is a top form given by the Mukai pairing h Ω ± , Ψ ± i ≡ (Ω ± ∧ λ (Ψ ± )) top , (2.21)where one defines the operator λλ (Ψ ± m ) ≡ ( − Int[ m/ Ψ ± m , (2.22)with Ψ m the degree m form in Ψ ± . The Mukai paring is invariant under B -transforms: h e ω Ω ± , e ω Ψ ± i = h Ω ± , Ψ ± i . For d = 6 the bilinear is anti-symmetric. The usual ac-tion of the exterior derivative on the component forms of Ω ± is compatible with thepatching (2.19) and defines an actiond : S ± ( E ) → S ∓ ( E ) , (2.23)while the generalized Lie derivative on spinors is given by L V Ω ± = L x Ω ± + d λ ∧ Ω ± = d( V · Ω ± ) + V · dΩ ± . (2.24)Note that given a splitting B the operator on Ω B ± ∈ S ± ( T X ⊕ T ∗ X ) corresponding tod is d H defined by d H Ω B ± ≡ e B d(e − B Ω B ± ) = (d − H ∧ ) Ω B ± , (2.25)7here H = d B . Furthermore one has L V Ω = e − B ( L V B − i x H ∧ ) Ω B , (2.26)where V B = e B V = x + ( λ − i x B ).Finally, we note that there is actually a slight subtlety in the relation betweengeneralized spinors and polyforms. Given the embedding (2.9) in O ( d, d ) of the GL ( d )action on the fibres of T X one actually finds that the Clifford action (2.16) impliesthat on U i we can identify S ± ( E ) with | Λ d T ∗ X | − / ⊗ Λ even/odd T ∗ X ; that is, there isan additional factor of the determinant bundle | Λ d T ∗ X | . (This factor is the source,for instance, of the fact that the Mukai pairing is a top form, rather than a scalar.)This bundle is trivial, so generalized spinors can indeed be written as polyforms patchedby (2.19), but there is no natural isomorphism to make this identification. The simplestsolution, and one which will also allow us to incorporate the dilaton in a natural way, isto extend the O ( d, d ) action to a conformal action O ( d, d ) × R + . One can then define afamily of spinor bundles S k ± ( E ) transforming with weight k under the conformal factor R + ; that is, with sections transforming as Ω ± → ρ k Ω ± where ρ ∈ R + . If one embedsthe GL ( d ) action on T X in O ( d, d ) as in (2.9) and, in addition, makes a conformalscaling by ρ = det a then sections of S − / ± ( E ) can be directly identified with polyformspatched by (2.19). A generalized metric G on E is the generalized geometrical equivalent of a Riemannianmetric on T X . We have seen that there is a natural O ( d, d ) structure on E defined bythe metric h· , ·i (2.5). The generalized metric G defines an O ( d ) × O ( d ) substructure.It splits E = C + ⊕ C − such that the metric h· , ·i gives a positive-definite metric on C + and a negative-definite metric on C − , corresponding to the two O ( d ) structure groups.One can define G as a product structure on E ; that is, G : E → E with G = 1 and h GU , GV i = h U, V i , so that (1 ± G ) project onto C ± . In general G has the form G = g − B g − g − Bg − B − Bg − ! = − B ! g − g ! B ! , (2.27)where g is a metric on X and B is a two-form. The patching of E implies B satis-fies (2.1), so that B may be identified with the curving of the gerbe used in the twistingof E . Thus the generalized metric G defines a particular splitting of E . In particular,8e see from (2.27) that G = e − B G e B where G is a generalized metric on T X ⊕ T ∗ X defined by g .The generalized metric G naturally encodes the NS fields g and B as the coset space O ( d, d ) /O ( d ) × O ( d ). The dilaton φ appears when one considers the conformal group O ( d, d ) × R + , used to define the generalized spinors as true polyforms. To define a O ( d ) × O ( d ) substructure in O ( d, d ) × R + , in addition to G which gives the embeddingin the O ( d, d ) factor, one must give the embedding ρ in the conformal factor ρ ∈ R + .Recall that under diffeomorphisms ρ transforms as a section of Λ d T X . Given the metric g we can define the generic embedding by ρ = e φ / √ g for some positive function e φ ,which we identify as the dilaton. Note that ρ is by definition invariant under O ( d, d )and so one finds the conventional T-duality transformation of the dilaton under O ( d, d ).Under the generalized Lie derivative, L V G = 0 implies [31] L x g = 0 , L x B − d λ = 0 , (2.28)so that L x H = 0 where H = d B . Such a V is called a generalized Killing vector .Given G we may decompose generalized spinors in Spin ( d, d ) under Spin ( d ) × Spin ( d ). In fact one can go further. Using the projection π : E → T X the two
Spin ( d ) groups can be identified and the generalized spinors may be decomposed asbispinors of Spin ( d ): Ω ± = e − φ e − B Φ even/odd . (2.29)In this expression, one first uses the Clifford map to identify the bispinors with ageneralized spinor Φ even/odd of S ± ( T X ⊕ T ∗ X ) ∼ = Λ even/odd T ∗ X and then uses thesplitting B to map to a spinor of S ± ( E ). The factor of e − φ appears because thepolyforms are really sections of S − / ± ( E ) transforming with weight − under conformalrescalings. Explicitly, if Φ is a bispinor, Φ ∈ Ω ∗ ( X ) a polyform, and γ i are Spin ( d )gamma matrices, the Clifford map is Φ = X k k ! Φ i ··· i k γ i ··· i k ←→ Φ = X k k ! Φ i ··· i k d x i ∧ · · · ∧ d x i k . (2.30)The Cliff( d, d ) action is realized via left and right multiplication by the gamma matrices γ i . For the chiral spinors Ω ± the sum is over k even/odd respectively. We also notehere the Fierz identity Φ = 1 n d X k k ! Tr ( Φ γ i k ··· i ) γ i ··· i k , (2.31)9here the γ i are n d × n d matrices. Finally, the generalized metric also defines an action ⋆ G on generalized spinors which is the analogue of the Hodge star. It is given byΩ ± ⋆ G Ω ± = e − B ⋆ λ (e B Ω ± ) , (2.32)where λ is the operator defined in (2.22) and ⋆ denotes the ordinary Hodge star for themetric g .If d = 2 n one can also introduce a generalized almost complex structure on E . Thisis a map J : E → E with J = − hJ U , J V i = h U, V i and gives a decomposition E C = L ⊕ ¯ L , (2.33)where L denotes the +i eigenspace of J . Note that L is maximally isotropic: h U, V i = hJ U , J V i = h i U , i V i = −h U, V i = 0. This defines a U ( n, n ) ⊂ O (2 n, n ) structure on E . By definition h U, J V i + hJ U , V i = 0, so J can be viewed either as an element of O (2 n, n ) or of the Lie algebra o (2 n, n ). A generic J can be written locally as J = I PQ − I ∗ ! , (2.34)where I ∗ is the linear map on T ∗ X dual to the map I on T X , P is a bivector and Q isa two-form. If the twisting (2.3) is trivial, so E = T X ⊕ T ∗ X , there are two canonicalexamples of generalized almost complex structures. The first is an ordinary almostcomplex structure I on T X , for which J = I − I ∗ ! . (2.35)The second is a non-degenerate (stable) two-form ω , for which J = ω − − ω ! . (2.36)If d ω = 0 this corresponds to a symplectic structure.More generally, a generalized almost complex structure J is integrable if L is closedunder the Courant bracket. That is, given U, V ∈ Γ( L ) then [ U, V ] ∈ Γ( L ). In theabove two cases (2.35), (2.36), this reduces to integrability of I and the closure of ω ,respectively. A generalized almost complex structure is equivalent to (the conformal Note that we have chosen the opposite sign in (2.35) compared with [29]. This is so that the +ieigenspace is identified with T (1 , ⊕ T ∗ (0 , . pure spinor Ω, which simply means a chiral complex generalized spinor suchthat the annihilator L Ω = { U ∈ E C : U · Ω = 0 } (2.37)is maximal isotropic. The sub-bundle L defined by J is then identified with L Ω .Notice that L Ω is invariant under conformal rescalings Ω f Ω, for any function f .A generalized almost complex structure is therefore more precisely equivalent to the pure spinor line bundle generated by Ω. Integrability of J can be expressed as thecondition dΩ = V · Ω for some V ∈ Γ( E ). If one can find a nowhere vanishing globallydefined Ω then one has an SU ( n, n ) structure and if in addition dΩ = 0 then one hasa generalized Calabi-Yau structure in the sense of [24]. For example, in the complexstructure case (2.35) one has Ω = c ¯Ω ( n, , where Ω ( n, is the holomorphic ( n, c is a non-zero constant (the reason why ¯Ω ( n, appears, rather than Ω ( n, , isdirectly related to the comment in footnote 1).A generalized vector V = x + λ is called (real) generalized holomorphic if L V J = 0.Equivalently, L V preserves the spinor line bundle generated by the corresponding purespinor Ω; that is, L V Ω = f Ω for some function f .Given a splitting B , one can define the corresponding generalized complex objectson T X ⊕ T ∗ X . In particular, if J is the generalized almost complex structure for a purespinor Ω, then the corresponding generalized almost complex structure on T X ⊕ T ∗ X is defined in terms of the annihilator ofΩ B = e B Ω (2.38)and is given by J B ≡ e B J e − B . (2.39)In particular, integrability of J is equivalently to integrability of J B using the twistedCourant bracket (2.15), or equivalently d H Ω B = V · Ω B .Viewing J as a Lie algebra element one can define its action on generalized spinorsvia the Clifford action [32]. Explicitly, one has J · = 12 (cid:16) Q mn d x m ∧ d x n ∧ + I mn [ i ∂ m , d x n ∧ ] + P mn i ∂ m i ∂ n (cid:17) . (2.40)Note that for any generalized vector V one has, under the Clifford action, [ J · , V · ] =( J V ) · . One can also define the operator J h : S ± ( E ) → S ± ( E ) J h ≡ e π J · , (2.41) Note that a different definition is used in [29]. J as an element of the group Spin ( d, d ). If n is even and the pure spinor is a section of S ± ( E ), then J h defines a complex structureon S ∓ ( E ), while if n is odd it defines a complex structure on S ± ( E ). Observe that forany generalized vector V we have the Clifford action identity J h · V · J − h · = ( J V ) · .Finally, a pair of generalized almost complex structures J and J are said to be compatible if [ J , J ] = 0 , (2.42)and the combination G = −J J (2.43)is a generalized metric. If Ω and Ω are the corresponding pure spinors, (2.42) isequivalent to J · Ω = J · Ω = 0. An example of a pair of compatible pure spinorsis (2.35), (2.36), with the compatibility condition being that I ki ω jk = g ij is positivedefinite. Note this is ω ij = − g ik I kj , this mathematics convention differing by a sign tothe usual physics convention. A pair of compatible almost complex structures definesan SU ( n ) × SU ( n ) structure. A generalized K¨ahler structure is an SU ( n ) × SU ( n )structure where both generalized almost complex structures are integrable, while for a generalized Hermitian structure only one need be integrable.Note that an SU ( n ) × SU ( n ) structure can equivalently be specified by a generalizedmetric and a pair of chiral Spin (2 n ) spinors. For example, for d = 6 a pair of chiralspinors η , η can be used to construct an SU (3) × SU (3) structure given byΩ + = e − φ e − B η ¯ η , Ω − = e − φ e − B η ¯ η − , (2.44)with η − ≡ ( η ) c . This will play a central role in the following sections. Similarly, for d = 4 a pair of chiral spinors η , η give rise to an SU (2) × SU (2) structure specifiedby two compatible pure spinors, but both of them consist of sums of even forms, sincenow ( η ) c is a positive chirality spinor. We will see such an SU (2) × SU (2) structurein section 4. That the spinors have the same chirality is necessary for them to becompatible in four dimensions [33]. 12 AdS backgrounds as generalized complex geome-tries AdS backgrounds Our starting point is the most general class of supersymmetric
AdS solutions of typeIIB supergravity, as studied in [19]. The ten-dimensional metric in Einstein frame is g E = e ( g AdS + g Y ) , (3.1)where g Y is a Riemannian metric on the compact five-manifold Y , and ∆ is a realfunction on Y . The AdS metric g AdS is normalized to have unit radius, so thatRic g AdS = − g AdS . (3.2)The ten-dimensional string frame metric is defined to be g σ ≡ e φ/ g E . In addition tothe metric, there is the dilaton φ and NS three-form H ≡ d B in the NS sector, andthe forms F ≡ F + F + F in the RR sector. The RR fluxes F n are related to the RRpotentials C n via F = d C , (3.3) F = d C − HC , (3.4) F = d C − H ∧ C . (3.5)These are all taken to be forms on Y , so as to preserve the SO (4 ,
2) symmetry, withthe exception of the self-dual five-form F which necessarily takes the form F = f (cid:16) vol AdS − f vol Y (cid:17) , (3.6)where f is a constant. Here f vol Y denotes a volume form for ( Y, g Y ). It is related by f vol Y = − vol to the volume form of [19], where the latter was given in terms of anorthonormal frame as vol = e and used to define, for instance, the Hodge star.In turns out that, in the Sasaki-Einstein limit, the conventional volume form is f vol Y rather than vol and so here we will use the former throughout. In particular, it is theorientation that we will use when defining integrals over Y .In [19] the conditions for a supersymmetric AdS background were written in termsof two five-dimensional spinors ξ , ξ on Y , giving the system of equations reproducedhere in (A.1)–(A.6) of appendix A. Various spinor bilinears involving ξ and ξ were13lso introduced, and used to determine the necessary and sufficient conditions forsupersymmetry. For example, it was shown that A ≡ (cid:0) ¯ ξ ξ + ¯ ξ ξ (cid:1) = 1 ,Z ≡ ¯ ξ ξ = 0 . (3.7)It will be useful for later in this paper to recall the definitions of the following scalarbilinears: sin ζ ≡ (cid:0) ¯ ξ ξ − ¯ ξ ξ (cid:1) ,S ≡ ¯ ξ c ξ , (3.8)the one-form bilinears: K ≡ ¯ ξ c β (1) ξ ,K ≡ ¯ ξ β (1) ξ ,K ≡ (cid:0) ¯ ξ β (1) ξ − ¯ ξ β (1) ξ (cid:1) ,K ≡ (cid:0) ¯ ξ β (1) ξ + ¯ ξ β (1) ξ (cid:1) , (3.9)and the two-form bilinears: V = − i2 ( ¯ ξ β (2) ξ − ¯ ξ β (2) ξ ) ,W = − ¯ ξ β (2) ξ . (3.10)Here the β m generate the Clifford algebra for g Y , so { β m , β n } = 2 g Y mn . Equivalently,with respect to any orthonormal frame, we write ˆ β m with { ˆ β m , ˆ β n } = 2 δ mn . We havealso introduced the notation β ( k ) ≡ k ! β m ··· m k d x m ∧ · · · ∧ d x m k .A key result of [19] is that K , the vector dual to the one-form K , defines a Killingvector that preserves all of the fluxes. This was identified as corresponding to the R -symmetry in the dual SCFT. Another important result wase − f = 4 sin ζ . (3.11)The Killing spinors ξ , ξ were used to introduce a canonical five-dimensional orthonor-mal frame in appendix B of [19], which is convenient for certain calculations. We willrefer to that paper for further details. Finally, we note that equation (A.22) of ap-pendix A may be used to obtain expressions for the two-form potentials B and C interms of the bilinear W introduced in (3.10): B = − f e φ/ Re W + b , (3.12) C = − f e φ/ C Re W − f e − φ/ Im W + c . (3.13)14ere b and c are real closed two-forms. Notice that the first term in B in (3.12) isa globally defined two-form, and thus H = d B is exact. It follows that [ H ] = 0 ∈ H ( Y, R ), although notice that b may be taken to be globally defined if and only if the torsion class of H is zero in H ( Y, Z ) (which for simplicity we shall assume). Similarremarks apply to C (up to large gauge transformations of C ). The supersymmetric
AdS geometry described above can be simply reformulated interms of generalized complex geometry, as in the discussion of [20, 21]. The basicobservation is simply that these solutions can be viewed as warped products of flatfour-dimensional space with a six-dimensional manifold X , satisfying a set of super-symmetry conditions that imply the existence of a particular generalized complex ge-ometry [22, 23]. As we shall explain in more detail below, combining this structurewith the existence of the Killing vector K precisely generalizes the correspondencebetween Sasaki-Einstein geometry and Calabi-Yau cone geometry. In the following weanalyze this reformulation in detail. We find in particular that all supersymmetric AdS solutions necessarily satisfy the condition of [20] that there is an SU (2)-structureon X .One begins by rewriting the unit AdS metric in a Poincar´e patch as g AdS = d r r + r g R , . (3.14)Switching to the string frame, we can consider (3.1) as a special case of a warpedsupersymmetric R , solution of the form g σ = e A g R , + g , (3.15)where the warp factor is given bye A = e φ/ r , (3.16)and the six-dimensional metric is given by g = e φ/ (cid:18) d r r + g Y (cid:19) . (3.17)We also define the six-dimensional volume form asvol ≡ e φ r d r ∧ f vol Y . (3.18)15otice that the six-dimensional manifold X , on which g is a metric, is a product R + × Y , where r may be interpreted as a coordinate on R + . In particular, X isnon-compact. It thus follows that supersymmetric AdS solutions are special cases ofsupersymmetric R , solutions.In [23] the general conditions for an N = 1 supersymmetric R , background, inthe string frame metric (3.15), were written in terms of two chiral six-dimensionalspinors η , η on X , namely the system of equations given here in (A.17)–(A.20). Therelation between the two sets of Killing spinors, for AdS solutions, is given by firstdecomposing Cliff(6) into Cliff(5) viaˆ γ m = ˆ β m ⊗ σ , m = 1 , . . . , γ = 1 ⊗ σ , (3.19)where ˆ γ i , i = 1 , . . . ,
6, generate Cliff(6) and σ α , α = 1 , ,
3, denote the Pauli matrices.Changing basis to ξ = χ + i χ , ξ = χ − i χ , we then have η = e A/ χ i χ ! , η − = e A/ − χ c i χ c ! η = e A/ − χ − i χ ! , η − = e A/ χ c − i χ c ! , (3.20)where χ ci ≡ ˜ D χ ∗ i denotes 5D charge conjugation, and correspondingly η i − ≡ ( η i + ) c ≡ D ( η i + ) ∗ where D = ˜ D ⊗ σ . For further details, see appendix A. Using the two chiralspinors η +1 , η +2 we may define the bispinorsΦ + ≡ η ⊗ ¯ η , Φ − ≡ η ⊗ ¯ η − . (3.21)Notice that, in the conventions of appendix A, we have A = 1, so that ¯ η ≡ η † isjust the Hermitian conjugate. Via the Clifford map (2.30) the bispinors for Spin (6) in(3.21) may also be viewed as elements of Ω ∗ ( X, C ). We will mainly tend to think ofΦ ± as complex differential forms of mixed degree. These are then Spin (6 ,
6) spinors, asexplained in section 2.2. In fact Φ ± in (3.21) are both pure spinors, and also compatible .They then define an SU (3) × SU (3) structure on T X ⊕ T ∗ X .In terms of (3.21), the Killing spinor equations for a general supersymmetric R , solution ( i.e. not necessarily associated with an AdS solution, but with vanishing16our-dimensional cosmological constant) may be rewritten as [34] (see also [22])d H (cid:0) e A − φ Φ − (cid:1) = 0 , (3.22)d H (cid:0) e A − φ Φ + (cid:1) = e A − φ d A ∧ ¯Φ + + 116 e A (cid:2) ( | a | − | b | ) F + i( | a | + | b | ) ⋆ λ ( F ) (cid:3) . (3.23)Here recall that F = F + F + F is the sum of RR fields and from (2.22) λ ( F ) = F − F + F . (3.24)Note that the Hodge star is with respect to the metric g , with positive orientationgiven by d r ∧ f vol Y . The remaining Bianchi identities and equations of motion are( cf . [34] equations (4.9)–(4.10))d H = 0 , d H F = δ source , (3.25)d(e A − φ ⋆ H ) − e A F n ∧ ⋆F n +2 = 0 , (3.26)(d + H ∧ )(e A ⋆ F ) = 0 . (3.27)The equation of motion for F can also be written asd (cid:2) e A e − B ⋆ λ ( F ) (cid:3) = 0 , (3.28)and follows from the supersymmetry equations. In fact, for AdS solutions it wasshown in [19] that supersymmetry implies all of the equations of motion and Bianchiidentities. We have also introduced the spinor norms | a | = | η +1 | , | b | = | η +2 | , (3.29)which for a supersymmetric R , background must satisfy | a | + | b | = e A c + , | a | − | b | = e − A c − , (3.30)where c ± are constants. Upon squaring and subtracting the equations one obtains | Φ ± | = 18 | a | | b | = 132 (cid:0) e A c − e − A c − (cid:1) . (3.31)As we now show, for the particular case of AdS solutions the above equationssimplify somewhat. In this case it is possible to fix the constant c − in (3.31) by thescaling of Φ ± with r which, using (3.16), implies that c − = 0 and hence | η +1 | = | η +2 | .17his is consistent with the equation Z = 0 in (3.7), since from (3.20) we see that | η ± | = | η ± | is equivalent to Re Z = 0. Notice that c − = 0 is also a necessary conditionin order to have supersymmetric probe branes [35]. The normalization that was usedin [19] implies | a | = | b | = e A and hence c + = 2. One can actually go a little further.In [20] it was assumed that there was an SU (2)-structure on the cone. In terms of thespinors η + i this is equivalent to the condition that, in addition to c − = 0, one has¯ η +1 η +2 + ¯ η +2 η +1 = 0 . (3.32)However it is easy to see that this is equivalent to Im Z = 0, which again is requiredby supersymmetry on Y . Thus in fact all supersymmetric AdS solutions necessarilysatisfy the SU (2) condition of [20].We now define the pure spinorΩ B − ≡ e A − φ Φ − , (3.33)which by (3.22) is d H closed, d H Ω B − = 0. The associated generalized almost complexstructure J B − is then integrable with respect to the twisted Courant bracket (2.15). Wealso define Ω − ≡ e − B Ω B − = e − B e A − φ Φ − , (3.34)which is closed under the usual exterior derivative:dΩ − = 0 . (3.35)The associated generalized almost complex structure, which we denote by J − , is thenintegrable. Combined with the fact that the norm of Φ − , and hence of Ω − , is nowherevanishing, this means, in particular, that we have a generalized Calabi-Yau manifoldin the sense of [24].We similarly define Ω + ≡ e − B e A − φ Φ + . (3.36)However, the corresponding generalized almost complex structure J + is not integrablein general, its integrability being obstructed by the RR fields in (3.23). If it were The generalized complex structures J B − and J − are related by (2.39). AdS solutions asdΩ − = 0 , dΩ + = d A ∧ ¯Ω + + i8 e A e − B ⋆ λ ( F ) . (3.37)It is worth noting that the latter equation may also be written asd (cid:0) e − A Re Ω + (cid:1) = 0 , (3.38)d (cid:0) e A Im Ω + (cid:1) = 18 e A e − B ⋆ λ ( F ) , (3.39)and that in turn equation (3.39) can be written as [36]e − B F = 8 J − · d (cid:0) e − A Im Ω + (cid:1) = 8d J − (cid:0) e − A Im Ω + (cid:1) , (3.40)where d J − ≡ − [d , J − · ]. For most of the paper we will demand that F = 0, orequivalently f = 0. Physically this corresponds to having non-vanishing D3-branecharge. It would be interesting to know whether or not all supersymmetric AdS solutions of type IIB supergravity have this property. In this section we examine the geometric properties of the generalized vector fields r∂ r , ξ ≡ J − ( r∂ r ) and η ≡ J − (d log r ). As in the Sasaki-Einstein case, r∂ r and ξ correspondrespectively to the dilatation symmetry and the R -symmetry in the dual SCFT (while η is related to a contact structure on Y , as we shall show later in section 6). We begin with the dilatation vector field r∂ r . It immediately follows from (3.16), (3.20)and (3.21) that L r∂ r Φ ± = Φ ± , (3.41)and therefore L r∂ r Ω ± = 3Ω ± . (3.42)This follows since e A has scaling dimension 2 (3.16), and both the B -field and thedilaton φ are pull-backs from Y . Notice that equation (3.42) may also be triviallyrewritten in terms of the generalized Lie derivative (2.24): L r∂ r Ω ± = 3Ω ± . (3.43)19his implies that L r∂ r J ± = 0 . (3.44)To see this, recall that J ± is defined by saying that its +i eigenspace is equal to the an-nihilator L Ω ± of Ω ± , and the latter is clearly preserved under the one-parameter familyof (generalized) diffeomorphisms generated by r∂ r . It further follows that L r∂ r G = 0,where G is the generalized metric G = −J + J − = −J + J − , so that r∂ r is generalizedKilling. Equation (3.44) says that r∂ r is a (real) generalized holomorphic vector fieldfor the integrable generalized complex structure J − . We shall not use this terminol-ogy for J + , since the latter is not in general integrable. Clearly, this generalizes theSasaki-Einstein result where the cone is Calabi-Yau and the dilatation vector r∂ r isholomorphic. R -symmetry We next define the generalized vectors ξ ≡ J − ( r∂ r ) , (3.45) η ≡ J − (d log r ) , (3.46)which are, in general, a mixture of vectors and one-forms. Recall that the generalizedalmost complex structures J ± are related to the generalized metric via G = −J + J − = −J − J + . The conical form (3.17) of the metric g and the fact that B has no componentalong d r implies that G d log r = e − − φ r∂ r , G r∂ r = e φ d log r , and hence inaddition to (3.46) we may also write ξ = e φ J + (d log r ) ,η = e − − φ J + ( r∂r ) . (3.47)We may split ξ and η into a vector part and a one-form part, in a fixed splitting of E , ξ = ξ v + ξ f , (3.48) η = η v + η f . (3.49)20y carrying out a calculation, presented in appendix B, we may then write these asbilinears constructed from the five-dimensional Killing spinors (3.9): ξ v = K ,ξ f = i ξ v b ,η v = e − − φ/ Re K ,η f = 4 f e K + i η v b . (3.50)As discussed in appendix B, it is the B -transform, ξ B , of the generalized vector ξ thatis naturally related to the bilinears of [19]. We have obtained (3.50) by performingan inverse B -transform using the expression for the B -field given in terms of bilinearspresented in (3.12). In particular, this is where the closed two-form b appears. Sincethe B -transform of b by an exact form is a generalized diffeomorphism, and a gaugesymmetry of string theory, we see that the physical information in b is represented byits cohomology class in H ( X, R ). More precisely, large gauge transformations of the B -field, which correspond to tensoring the underlying gerbe by a unitary line bundleon X , lead to the torus H ( X, R ) /H ( X, Z ) (with suitable normalization). Turningon the two-form b corresponds to giving vacuum expectation values to moduli (of theNS field B ) and so is a symmetry of the supersymmetry equations. It is therefore leftundetermined. In the field theory dual, the cohomology class of b thus corresponds toa marginal deformation.In [19] it was shown that K is a Killing vector that preserved all of the fluxes,and thus K was identified as being dual to the R -symmetry in the SCFT. In thegeneralized geometry we can show the stronger conditions that L ξ J ± = 0 , (3.51)and hence ξ is a generalized holomorphic Killing vector field. In fact it is straight-forward to show L ξ Ω − = − − and hence L ξ J − = 0. Indeed since dΩ − = 0 and r∂ r − i ξ ∈ L Ω − annihilates Ω − , using (2.24) and (3.43) we have L ξ Ω − = d ( ξ · Ω − ) = − id ( r∂ r · Ω − ) = − i L r∂ r Ω − = − − . (3.52)In appendix B we show that L ξ Ω + = 0 and hence L ξ J + = 0. There we also show that L ξ (e − B F ) = 0 . (3.53)Thus, we have established that ξ ≡ J − ( r∂ r ) is a generalized holomorphic vector field,which moreover is generalized Killing for the generalized metric G = −J − J + , and21lso preserves the RR fluxes. Again, this clearly generalizes the Sasaki-Einstein result,where ξ = I ( r∂ r ) is a holomorphic Killing vector field for the Calabi-Yau cone.To conclude this section we note that when f = 0 the vector field ξ v = K isnowhere vanishing on Y = { r = 1 } . One can see this from the formula | K | = sin ζ + | S | , (3.54)and using (3.11). Thus for f = 0, ξ v acts locally freely on Y and hence the orbits of ξ v define a corresponding one-dimensional foliation of Y . This is again precisely as in theSasaki-Einstein case (although in the Sasaki-Einstein case the norm of ξ v is constant). AdS backgrounds Recall that in the Sasaki-Einstein case one can consider the symplectic reduction of theCalabi-Yau cone metric with respect to the R -symmetry Killing vector ξ (or equiva-lently a holomorphic quotient with respect to r∂ r − i ξ ). Generically ξ does not define a U (1) fibration and the four-dimensional reduced space is not a manifold. Nonetheless,locally one can consider the geometry on the transversal section to the foliation formedby the orbits of ξ in the Sasaki-Einstein space. The result of the reduction is that thisfour-dimensional geometry is K¨ahler-Einstein. Thus locally one can always write theSasaki-Einstein metric as g Y = η ⊗ η + g KE (4.1)where g KE is a K¨ahler-Einstein metric.The existence of the generalized holomorphic vectors ξ and r∂ r in the generic casesuggests one can make an analogous generalized reduction to four dimensions. Inthis section, we show that this is indeed the case following the theory of generalizedquotients developed in [26, 27]. We first review the formalism and then apply it to ourparticular case, showing that there is a generalized Hermitian structure on the localtransversal section, giving the conditions satisfied by the corresponding reduced purespinors. We will follow the description of generalized quotients given in [27] . These includeboth symplectic reductions and complex quotients as special cases. One first needs to Note that the bracket [[ , ]] used in [27] is the Dorfman bracket or generalized Lie derivative [[ V, W ]] = L V W and is not anti-symmetric. reduction data .In conventional geometry, the action of a Lie group G on M is generated infinitesi-mally by a set of vector fields, defined by a map from the Lie algebra ψ : g → Γ( T M ).Given a vector field x ∈ Γ( T M ), the infinitesimal action of u ∈ g is then just the Liederivative (or in this case Lie bracket) δx = L ψ ( u ) x = [ ψ ( u ) , x ] . (4.2)One requires that given u, v ∈ g , one has [ ψ ( u ) , ψ ( v )] = ψ ([ u, v ]) so that L ψ ( u ) L ψ ( v ) − L ψ ( v ) L ψ ( u ) = L [ ψ ( u ) ,ψ ( v )] = L ψ ([ u,v ]) , (4.3)and thus there is a Lie algebra homomorphism between g and the algebra of vectorfields under the Lie bracket.In generalized geometry, we have a larger group of symmetries, diffeomorphisms and B -shifts, which are generated infinitesimally by the generalized Lie derivative (2.14).Thus given an action of G on M , it is natural to consider the infinitesimal “liftedaction” of G on E defined by the map ˜ ψ : g → Γ( E ), such that for any V ∈ Γ( E ) and u ∈ g we have δV = L ˜ ψ ( u ) V , (4.4)and under the projection π : E → T M we simply get the vector fields ψ ( u ), that is π ˜ ψ ( u ) = ψ ( u ) . (4.5)Such transformations are infinitesimal automorphisms of E that have the property thatthey preserve both the metric h· , ·i on E and the Courant bracket (2.13). If we againassume that L ˜ ψ ( u ) L ˜ ψ ( v ) − L ˜ ψ ( v ) L ˜ ψ ( u ) = L ˜ ψ ([ u,v ]) , (4.6)then ˜ ψ defines an equivariant structure on E . (Note that this is equivalent to theCourant bracket condition [ ˜ ψ ( u ) , ˜ ψ ( v )] = ˜ ψ ([ u, v ]).) In what follows it will also beassumed that ˜ ψ is isotropic, that is h ˜ ψ ( u ) , ˜ ψ ( u ) i = 0 (4.7)for all u , u ∈ g .One can actually define a more general action on E which is a homomorphismbetween algebras with Courant brackets rather than Lie algebras. One starts by ex-tending g to a larger algebra. The construction considered in [27] which is relevant for23s is as follows. Let h be a vector space on which there is some representation of g .Then we can form a “Courant algebra” a = g ⊕ h with Courant bracket given u i ∈ g and w i ∈ h [( u , w ) , ( u , w )] = ( [ u , u ] , ( u · w − u · w ) ) , (4.8)where u · w is the action of g on h . Suppose in addition µ : M → h ∗ is a g -equivariantmap, meaning L ψ ( u ) µ ( w ) = µ ( u · w ) for all u ∈ g and w ∈ h . Then, given some isotropiclifted action ˜ ψ of g , one can then define the extended action Ψ : g ⊕ h → Γ( E ) , ( u, w ) ˜ ψ ( u ) + d µ ( w ) , (4.9)which, it is easy to show, has the property that[Ψ( u , w ) , Ψ( u , w )] = Ψ([ u , u ] , ( u · w − u · w )) . (4.10)and hence Ψ defines a homomorphism of Courant algebras as opposed to Lie algebrasas in (4.3). Note that the extra factor d µ ( w ) in (4.9) corresponds to a trivial B -shift,and thus L Ψ( u,v ) = L ˜ ψ ( u ) . Note also that µ will play the role of a moment map. In theconventional case of symplectic reductions one has µ : M → g ∗ , whereas here h can beany representation space. The triple ( ˜ ψ, h , µ ) is known as the reduction data .This reduction data can then be used to define a reduced generalized tangent bundle E red . First one makes the usual assumptions about µ and the G action on M so that M red = µ − (0) /G is a manifold. (This requires that 0 is a regular point of µ and thatthe G action on µ − (0) is free and proper.) Then define the sub-bundle K which is theimage of the bundle map a × M associated to Ψ, that is K = n ˜ ψ ( u ) + d µ ( w ) , u ∈ g , w ∈ h o ⊆ E , (4.11)and also the orthogonal bundle K ⊥ , the fibres of which are orthogonal to K withrespect to the O ( d, d ) metric h· , ·i . One can then construct the generalized tangentspace on M red E red = K ⊥ | µ − (0) K | µ − (0) (cid:30) G . (4.12)The main results of [26, 27] are then to show how various geometrical structures can betransported from E to E red . The case of particular interest to us is that of generalizedHermitian reduction. Note we take a slightly different definition of the bracket to that in [27] in to order to match theCourant bracket (2.13) on E .
24s discussed in section 2.2 a generalized Hermitian manifold is a generalized complexmanifold with a compatible generalized metric (or equivalently a second, compatible,generalized almost complex structure). Let J be the integrable generalized complexstructure and G be the generalized metric. Given some reduction data ( ˜ ψ, h , µ ), recall-ing L Ψ( u,v ) = L ˜ ψ ( u ) , the structures are G -invariant if L ˜ ψ ( u ) G = L ˜ ψ ( u ) J = 0 , (4.13)for all u ∈ g . One can also define the sub-bundles K G = GK ⊥ ∩ K ⊥ , (4.14)which is the sub-bundle of K ⊥ the fibres of which are orthogonal to K , with respectto G , and E K = K ⊕ GK (4.15)which is the G -orthogonal complement to K G . Theorem 4.4 of [27] then states Theorem 1 (Generalized Hermitian reduction [27])
Let E be a generalized tan-gent space over M with reduction data ( ˜ ψ, h , µ ) . Suppose E is equipped with a G -invariant generalized Hermitian structure ( J , G ) . If over µ − (0) , J K G = K G , orequivalently J E K = E K , then J and G can be reduced to E red where they define ageneralized Hermitian structure. Even if the group action is such that the reduced space is not a manifold, one can stilldefine a generalized Hermitian structure on the transversal section to the foliation. ξ We now use the reduction formalism to show that the generalized Calabi-Yau geometryon the cone X reduces to a generalized Hermitian geometry in four dimensions. Thisis the analogue of the reduced K¨ahler-Einstein geometry in the Sasaki-Einstein case.First we note that there is a group action on the cone X generated by the vectors r∂ r and ξ v . These commute and the corresponding Lie algebra is simply R ⊕ R . If theorbits of ξ v form a U (1) action then together r∂ r and ξ v integrate to a C ∗ action, butthis need not be the case. The generalized vectors r∂ r and ξ give a lifted action of R ⊕ R on E , so that, if u = ( a, b ) ∈ R ⊕ R ,˜ ψ ( u ) = ar∂ r + bξ . (4.16)25y definition we have π ˜ ψ ( u ) = ar∂ r + bξ v . Under the Courant bracket, given theexpressions (3.50) we see that [ r∂ r , ξ ] = 0, and hence[ ˜ ψ ( u ) , ˜ ψ ( u )] = 0 = ˜ ψ ([ u , u ]) (4.17)for all u and u , as required for a lifted action. Furthermore, from (B.5) we see that˜ ψ is isotropic.We also have a generalized Hermitian structure on X given by J ± . The generalizedcomplex structure J − is integrable, and we have the compatible generalized metric G = −J + J − . In section 3 we showed L r∂ r J ± = L ξ J ± = 0 and hence the Hermitianstructure is invariant under both group actions.There are then two different ways we can view the generalized reduction, mirroringthe symplectic reduction and the complex quotient in the Sasaki-Einstein case. In thefirst reduction, we take g = R generated by ξ v , and in the reduction data we take h = g and µ = log r . This is the same moment map one takes in the symplectic reduction.In the second case we take the complex Lie algebra g = C generated by r∂ r − i ξ v and h = 0 so there is no moment map. The reduction is then analogous to a complexquotient. We now discuss these in turn. As in the Sasaki-Einstein case, both lead tothe same reduced structure. g = R reduction In this case, the reduction data is˜ ψ ( u ) = uξ , h = R , µ = log r . (4.18)We have already seen that ˜ ψ is an isotropic lifted action. It is also clear that µ is g -equivariant since, from (B.5), i ξ v d µ = 0. Thus ( ˜ ψ, h , µ ) are suitable reduction data.Furthermore, J − and G are both invariant under ˜ ψ ( u ). We have µ − (0) = Y , (4.19)and given u ∈ g and v ∈ h K = { uξ + v d log r } . (4.20)Using Gξ = e φ/ η , G d log r = e − − φ/ r∂ r (4.21)we have GK = { u ′ η + v ′ r∂ r } , (4.22)26nd hence E K ≡ K ⊕ GK = { uξ + v d log r + u ′ η + v ′ r∂ r } . (4.23)Using the definitions (3.46) we immediately see that J − E K = E K . Hence, assumingthe action of ξ v on Y gives a U (1) fibration, using the generalized Hermitian reductiontheorem, we see that we have a generalized Hermitian structure on E red over the four-dimensional space M red = Y /U (1). More generally, we get a generalized Hermitianstructure on the transversal section to the ξ v orbits. g = C reduction In this case, the reduction data is˜ ψ ( u ) = u ( r∂ r − i ξ ) , h = 0 , µ = 0 . (4.24)Given h is trivial, we have K = { u ( r∂ r − i ξ ) } . (4.25)As before, we have already seen that ˜ ψ is an isotropic lifted action and so ( ˜ ψ, h , µ )are suitable reduction data. J − and G are both invariant under ˜ ψ ( u ) and finally,using (4.21), we now have GK = { u ′ (d log r − i η ) } , (4.26)and hence E K = { u ( r∂ r − i ξ ) + u ′ (d log r − i η ) } , (4.27)Again we immediately see that J − E K = E K . Hence, again assuming the action of ξ v on Y gives a U (1) fibration, using the generalized Hermitian reduction theorem, we seethat we have a generalized Hermitian structure on E red over the four-dimensional space M red = X/ C ∗ = Y /U (1), or more generally, we get a generalized Hermitian structureon the transversal section to the r∂ r − i ξ v orbits.Note that in both cases the reduced manifold M red is the same. Furthermore, the(complexified) spaces E K , and hence the G -orthogonal complements K G , also agree. Asdiscussed in [27], K G is a model for the reduced bundle E red . Thus the two reductionsgive identical generalized Hermitian structures on M red . We now calculate the conditions on the reduced generalized Hermitian structure impliedby supersymmetry. The reduced structure can be defined by a pair of commuting27eneralized almost complex structures: ˜ J which is integrable and is the reductionof J − , and a non-integrable structure ˜ J , defined such that − ˜ J ˜ J is the reducedgeneralized metric. Equivalently, the structures are defined as a pair of pure spinors˜Ω and ˜Ω . It is the differential conditions on ˜Ω and ˜Ω implied by supersymmetrythat we will derive.In order to construct the reduced pure spinors, first note that the reduction gives asplitting of the generalized tangent space E = E K ⊕ K G (4.28)such that, in general, the O ( d, d ) metric h· , ·i factors into an O ( p, p ) metric on K G andan O ( d − p, d − p ) metric on E K . Thus we can similarly decompose sections of thespinor bundles S ± ( E ) into spinors of Spin ( d − p, d − p ) × Spin ( p, p ). In particular,generic sections Ω ± ∈ S ± ( E ) can be written asΩ ± = Θ ± ⊗ ˜Ω + ⊕ Θ ∓ ⊗ ˜Ω − . (4.29)It is then the spinor components of ˜Ω ± in S ± ( K G ) which correspond to the reducedpure spinors. For the case in hand the relevant decomposition is under Spin (2 , × Spin (4 , ⊂ Spin (6 , − = Θ − ⊗ ˜Ω , Ω + = Θ + ⊗ ˜Ω . (4.30)Thus the reduced spinors ˜Ω and ˜Ω are both positive helicity in Spin (4 , Spin (6 ,
6) gamma matrices reflectingthe decomposition (4.28). We first introduce coordinates adapted to the reduction. Wewrite the R -symmetry Killing vector as ξ v = K = ∂ ψ , (4.31)Let y m be coordinates on the transversal section to the R -symmetry foliation. Thismeans that i ξ v d y m = 0 and, in particular, the metric decomposes as g Y = K ⊗ K + g red mn d y m d y n , (4.32) Note that this, more conventional, normalization of ψ differs from the corresponding coordinatein [19] by a factor of three.
28n analogy to (4.1). The reduction structure already defines a natural basis on E K given by ˆ f = r∂ r , f = d log r , ˆ f = ξ , f = η , (4.33)and satisfying h f i , ˆ f j i = δ ij and h f i , f j i = h ˆ f i , ˆ f j i = 0. We can then define anorthogonal basis on K G given byˆ e m = e − b ∂ y m − ˜ η m ξ , e m = d y m − η m ξ (4.34)where ˜ η m = 2 h η, e − b ∂ y m i and η m = 2 h η, d y m i = i η v d y m . This basis again satisfies h e m , ˆ e n i = δ mn and h e m , e n i = h ˆ e m , ˆ e n i = 0.Given such a basis we can then write a generic Spin (6 ,
6) spinor using the standardraising and lowering operator construction. Consider the polyform Ω (0) = e − b ∈ S + ( E ). It is easy to see that we have the Clifford actionsˆ f i · Ω (0) = ˆ e m · Ω (0) = 0 , (4.35)for all i and m . Thus we can regard Ω (0) as a ground state for the lowering operators( ˆ f i , ˆ e m ). A generic spinor is then given by acting with the anti-commuting raisingoperators ( f i , e m ). Acting with the e m first, we see that a generic (non-chiral) spinorhas the form Ω = e − b ˜Ω + f · e − b ˜Ω + f · e − b ˜Ω + f · f · e − b ˜Ω , (4.36)where ˜Ω i are polyforms in d y m , and e − b ˜Ω i transform as a Spin (4 ,
4) spinor under theClifford action of ( e m , ˆ e m ).We can now write the supersymmetry pure spinors Ω ± in the form (4.36). Requiringthat r∂ r − i ξ and d log r − i η annihilate Ω − while r∂ r − ie φ/ η and d log r − ie − − φ/ ξ annihilate Ω + one finds the only possibility isΩ − = (d log r − i η ) · r e − ψ e − b ˜Ω , Ω + = (cid:0) φ/ d log r · η (cid:1) · r e − b ˜Ω , where ˜Ω and ˜Ω are both even polyforms in d y m , as claimed in (4.30). We haveintroduced factors of r and e − ψ so that ˜Ω and ˜Ω are independent of the r and ψ coordinates. In general, they are then only locally defined.One can then derive the conditions on ˜Ω and ˜Ω , reduced to the transversal section,implied by supersymmetry. From dΩ − = 0 one findsd ˜Ω = − η · ˜Ω , (4.37)29here ˜ η is generalized vector on the transversal section defined by η = d ψ + e − b ˜ η . (4.38)This is means that ˜Ω defines an integrable generalized complex structure on the trans-verse section, as expected from the reduction theorem.For the second pure spinor, the condition (3.38) on Re Ω + is equivalent tod (cid:0) e ∆+ φ/ ˜ η · Im ˜Ω (cid:1) = − − ∆ − φ/ Re ˜Ω , d (cid:0) e ∆+ φ/ Im ˜Ω (cid:1) = 0 . (4.39)Finally, since i r∂ r F = 0, following (4.36), we can decompose the flux ase − B F = e − b ˜ F + η · e − b ˜ G. (4.40)The final condition (3.40) is then equivalent tod (cid:0) e − ∆ − φ/ ˜ η · Re ˜Ω (cid:1) = − ˜ G , (cid:2) ˜ J d(4∆ + φ ) (cid:3) · Im ˜Ω = − e φ/ ˜ F , (4.41)and to ˜ J · d (cid:0) e − ∆ − φ/ ˜ η · Re ˜Ω (cid:1) = 0 , (4.42)where ˜ J is the reduction to the transverse section of the generalized complex structure J b − = e b J − e − b , and we have used the compatibility relation ˜ J · ˜Ω = 0. The conditions (4.39) and (4.42), which do not involve the flux, can be viewed asa generalization of the usual K¨ahler-Einstein conditions. Given an ˜Ω satisfying theseconditions, the flux is then determined by (4.41). Ω − The closed pure spinor Ω − is associated with the integrable generalized complex-structure J − . The latter in turn holds information regarding BPS operators in thedual field theory. In this section we explore two aspects of this duality. The first isthe mesonic moduli space of the dual theory, which is known to correspond to the sub-space for which the polyform Ω − reduces to a three-form. The second is the connectionbetween generalized holomorphic objects and dual BPS operators. Note that in the language defined in section 5.3 below, the condition (4.42) states that d (cid:0) e − ∆ − φ/ ˜ η · Re ˜Ω (cid:1) (and hence ˜ G ) is an element of U J . .1 The general form of Ω − Recall that the most general pure spinor takes the form [29]Ω = αθ ∧ θ ∧ ... ∧ θ k ∧ e − b +i ω , (5.1)where α is some complex function, θ i are complex one-forms, while b and ω are bothreal two-forms. The integer k is called the type of the pure spinor, which can changealong various subspaces of X .Using the definition of Ω ± , the Fierz identity (2.31), and the results of section 3.1and appendix A, one can find expressions for Ω ± in terms of spinor bilinears introducedin [19]. We find that in general Ω − is of type one withΩ − = θ ∧ e − b − +i ω − , (5.2)where θ = − r (i K + S d log r ) ,ω − = 4e φ/ f (sin 2 ¯ φ ) (cid:0) K ∧ Im( K ) − cos 2¯ θ cos 2 ¯ φ Re( K ) ∧ d log r (cid:1) ,b − = − φ/ f (sin 2 ¯ φ ) (cid:0) K ∧ Re( K ) + (cos 2 ¯ φ ) Im( K ) ∧ d log r (cid:1) + b . (5.3)Note that ω − and b − are not uniquely defined since we can add two-forms that vanishwhen wedged with θ . Here the angles ¯ θ and ¯ φ , which appear in appendix B of [19]without bars, are functions on the link Y that are related to the scalar spinor bilinearsthrough sin ζ = cos 2¯ θ cos 2 ¯ φ , (5.4) | S | = − sin 2¯ θ cos 2 ¯ φ . (5.5)Using the results of [19], we have the important result that θ is exact θ = d (cid:20) −
124 e r S (cid:21) ≡ d (cid:0) r θ (cid:1) . (5.6)Alternatively, from the supersymmetry equation dΩ − = 0 and the definite scalingdimension L r∂ r Ω − = 3Ω − , we immediately obtainΩ − = d( r∂ r y Ω − ) , (5.7)the one-form part of which reduces to (5.6). The fact that θ is closed was essentially observed in [37], and it was also shown to be exact in thespecial cases of the Pilch-Warner and Lunin-Maldacena solutions in [20]. .2 Type change of Ω − and the mesonic moduli space The pure spinor Ω − has the property that its type can jump from type one to typethree on the locus θ = 0. This locus can be neatly parameterized through the angles ¯ θ and ¯ φ . Assuming f = 0, we have from (3.11) that sin ζ is nowhere zero and then (5.4)implies that both cos 2 ¯ φ and cos 2¯ θ are nowhere zero. Using the expression for K inappendix B of [19], we see that when f = 0sin 2¯ θ = 0 ⇐⇒ θ = 0 , (5.8)sin 2¯ θ = sin 2 ¯ φ = 0 ⇐⇒ θ = 0 . (5.9)The locus θ = 0 is thus a sublocus of θ = 0. Notice that, where θ = 0, Ω − is notidentically zero, as one might have naively expected from (5.2), but instead reduces toa finite, non-zero three-form. Indeed, the powers of sin 2 ¯ φ in the denominator of b − and ω − are cancelled by those in K , K and K .The locus θ = 0 is precisely where a probe pointlike D3-brane in X is supersymmet-ric. This follows from [37] where it was shown that the pull-back of θ to the D3-braneworldvolume is equal to the F-term of the worldvolume theory. The supersymmetriclocus of such a pointlike D3-brane is naturally interpreted as the mesonic moduli space. In the Sasaki-Einstein case, holomorphic functions on the Calabi-Yau cone with adefinite scaling weight λ under the action of r∂ r also have a charge λ under the action of ξ . This stems from the intimate connection (via Kaluza-Klein reduction on the Sasaki-Einstein manifold) between holomorphic functions on the cone and BPS operators inthe dual CFT, in fact (anti-)chiral primary operators. For general AdS solutions wemight expect that the holomorphic functions should be replaced by polyforms and thatthe BPS condition of matching charges should be with respect to the generalized Liederivative L discussed in section 2. We now derive such a result, leaving the detailedconnection with Kaluza-Klein reduction on the internal space Y to future work.We first recall that a generalized almost complex structure J defines a grading ongeneralized spinors, or equivalently differential forms. If Ω ∈ Γ( S ± ( E )) is a pure spinorcorresponding to J , one defines the canonical pure spinor line bundle U n J ⊂ S ± ( E ) assections of the form ϕ = f Ω for some function f . One can then define U ( n − k ) J = ∧ k ¯ L ⊗ U n J . (5.10)32lements of U k J have eigenvalues i k under the Lie algebra action of J given in (2.40).These bundles then give a grading of the spinor bundles S ± ( E ). A generalized vector V ∈ Γ( E ) acting on an element of U k J gives an element of U k +1 J ⊕ U k − J . In particularan annihilator of Ω acts by purely raising the level by one. If the generalized complexstructure J is also integrable then the exterior derivative splits into the sumd = ∂ J + ¯ ∂ J , (5.11)where C ∞ (cid:0) U k J (cid:1) ¯ ∂ J ←−−→ ∂ J C ∞ (cid:0) U k − J (cid:1) . (5.12)Consider now a spinor ψ satisfying ψ ∈ U k J − , L r∂ r ψ = λψ , (5.13)for some k and λ . Then imposing in addition¯ ∂ J − ψ = 0 , ( r∂ r + i ξ ) · ψ = 0 implies L ξ ψ = i L r∂ r ψ . (5.14)In other words, subject to the constraints (5.13), a spinor is BPS if it is generalizedholomorphic and is annihilated by r∂ r + i ξ . To see this result, we first write r∂ r =(1 / r∂ r + i ξ ) + (1 / r∂ r − i ξ ) and use (5.13) to deduce that ∂ J − [( r∂ r + i ξ ) · ψ ] + ( r∂ r + i ξ ) · ∂ J − ψ = 0 , ¯ ∂ J − [( ∂ r − i ξ ) · ψ ] + ( r∂ r − i ξ ) · ¯ ∂ J − ψ = 0 . (5.15)In obtaining this we used the fact that since r∂ r − i ξ is an annihilator of Ω − it raisesthe level of ψ and similarly r∂ r + i ξ lowers the level. We then compute L ξ ψ = i L r∂ r ψ − i { d [( r∂ r + i ξ ) · ψ ] + ( r∂ r + i ξ ) · d ψ } = i L r∂ r ψ − i (cid:8) ¯ ∂ J − [( r∂ r + i ξ ) · ψ ] + ( r∂ r + i ξ ) · ¯ ∂ J − ψ (cid:9) . (5.16)In a similar way, given (5.13) we also have ∂ J − ψ = 0 , ( r∂ r − i ξ ) · ψ = 0 implies L ξ ψ = − i L r∂ r ψ . (5.17)33 The pure spinor Ω + Ω + One can see immediately from the supersymmetry equation (3.39) that if we assume F = 0, which we shall do, then Im Ω + must have a scalar component and hence Ω + isof type 0: Ω + = α + e − b + +i ω . (6.1)Using the same procedure as in the last section, we may again express these quantitiesin terms of the bilinears of [19]. After defining the rescaled two-form ω = e − A r ω , (6.2)we find α + = − i32 f e − A r ,ω = − r f e ( V + K ∧ d log r ) ,b + = e φ/ f Im K ∧ d log r + b , (6.3)where b appears in (3.12). The rescaling (6.2) is motivated by the fact that ω defines a canonical symplecticstructure. To see this, we first observe that Y admits a contact structure defined bythe one-form σ ≡ f e K . (6.4)Recall that for a one-form σ to be contact, the top-degree form σ ∧ d σ ∧ d σ must benowhere vanishing. Using (3.19) of [19], and results in appendix B of [19], one caneasily show that σ ∧ d σ ∧ d σ = 128 f e f vol Y = 8sin ζ f vol Y , (6.5)where recall f vol Y = − e (using the orthonormal frame in appendix B of [19]). Wethen observe, using (3.19) of [19], that ω = d( r σ ) , (6.6)34hich shows that ω is closed and non-degenerate, and hence defines a symplectic struc-ture on the cone X = R + × Y . Alternatively, one can see the formula (6.6) for ω directlyfrom the supersymmetry equation (3.38) on noting that e − A Ω + has scaling dimension2 under r∂ r . Furthermore, again using the results of appendix B of [19], we have1 = ξ v y σ , ξ v y d σ , (6.7)which shows that ξ v is also the unique “Reeb vector field” associated with the contactstructure. Notice also that (6.6) implies that H = r / ξ v , i.e. d H = − i ξ v ω . It is remarkable thatthese features, which are well-known in the Sasaki-Einstein case, are valid for all su-persymmetric AdS solutions with non-vanishing five-form flux.Although we have a symplectic structure, we do not quite have a K¨ahler structure,as in the Calabi-Yau case, but it is quite close. Using the last equation in (3.50) andthe definition (6.4) we see that η f = σ + i η v b , (6.8)and thus (cid:0) e b η (cid:1) | − form = σ . Since e b (d log r ) = d log r manifestly, and by definition η ≡ J − (d log r ), we have, using (2.34), σ = J b − (d log r ) | − form = − ( I b − ) ∗ (d log r ) . (6.9)Note this is precisely analogous to the formula for the contact form in the Sasakiancase. We then haved J b − r = − r d (cid:18) Q b − + 12 Tr I b − (cid:19) − ( I b − ) ∗ (d( r )) , (6.10)where here we recall that in general we define d J − ≡ − [d , J − ], and we use (2.40) forthe action on generalized spinors. From this it follows that ω = 14 dd J b − r + 14 d( r ) ∧ d (cid:18) Q b − + 12 Tr I b − (cid:19) . (6.11)Thus r is almost a K¨ahler potential, for the b -transformed complex structure J b − =e b J − e − b , except for the last term. Recall that in any four-dimensional CFT there are two central charges, usually called a and c , that are constant coefficients in the conformal anomaly h T µµ i = 1120(4 π ) (cid:16) c (Weyl) − a (cid:17) . (6.12)35ere T µν denotes the stress-energy tensor, and Weyl and Euler denote certain curvatureinvariants for the background four-dimensional metric. For SCFTs, both a and c arerelated to the R -symmetry [8] via a = 332 (cid:0) R − Tr R (cid:1) , c = 132 (cid:0) R − R (cid:1) . (6.13)Here the trace is over the fermions in the theory. For SCFTs with AdS gravity duals,in fact a = c holds necessarily in the large N limit [38]. The central charge of theSCFT is then inversely proportional to the dual five-dimensional Newton constant G [38], obtained here by Kaluza-Klein reduction on Y . The Newton constant, in turn,was computed in appendix E of [19], and is given by G = G V = κ πV , (6.14)where G is the ten-dimensional Newton constant of type IIB supergravity, and wehave defined V ≡ Z Y e f vol Y . (6.15)We may derive an alternative formula for G as follows. We begin by rewriting V = f Z Y ζ f vol Y , (6.16)where we have used the relation (3.11). Importantly, the constant f is quantized, beingessentially the number of D3-branes N . Specifically, we have N = 1(2 πl s ) g s Z Y d C = 1(2 πl s ) g s Z Y ( F + H ∧ C ) . (6.17)Using the Bianchi identity DG = − P ∧ G ∗ and the result (A.22), one derives thatd( H ∧ C ) = − (2 /f )d[e Im( W ∗ ∧ G )] and so we can also write N = 1(2 πl s ) g s Z Y (cid:18) F − f Im [ W ∗ ∧ G ] (cid:19) . (6.18)We may evaluate this expression in terms of the orthonormal basis of forms e i intro-duced in appendix B of [19], and after some calculation we find N = − f (2 πl s ) g s Z Y ζ f vol Y . (6.19)36ombining these formulae and using 2 κ = (2 π ) l s g s leads to the result G = 8 V π f N . (6.20)Consider now the integral µ = 1(2 π ) Z X e − r / ω . (6.21)This is the Duistermaat-Heckman integral for a symplectic manifold ( X, ω ) with Hamil-tonian function H = r /
2, which we have shown is the Hamiltonian for the Reeb vectorfield ξ v . Using (6.6) and (6.5) we may rewrite ω
3! = 16 f e r d r ∧ f vol Y . (6.22)Performing the r -integral in (6.21) allows us to rewrite the five-dimensional Newtonconstant as G = πµ N . (6.23)Since µ = 1 for the round five-sphere solution, we thus obtain the ratio G G S = µ .Recalling that this is, by AdS/CFT duality, the inverse ratio of central charges [38],we deduce the key result a N =4 a = 1(2 π ) Z X e − r / ω
3! = 1(2 π ) Z Y σ ∧ d σ ∧ d σ . (6.24)Here a N =4 = N / N ) central charge of N = 4 super-Yang-Millstheory.The formula (6.24) implies that the central charge depends only on the symplecticstructure of the cone ( X, ω ) and the Reeb vector field ξ v . This is perhaps surprising:one might have anticipated that the quantum numbers of quantized fluxes would appearexplicitly in the central charge formula. However, recall from formulae (3.12), (3.13)that the two-form potentials B and C are globally defined. In particular, for example,the period of H = d B through any three-cycle in Y is zero.As discussed in [11], the Duistermaat-Heckman integral in (6.24) may be evaluatedby localization. The integral localizes where ξ v = 0, which is formally at the tip of thecone r = 0. Unless the differentiable and symplectic structure is smooth here (whichis only the case when X ∪ { r = 0 } is diffeomorphic to R ), one needs to equivariantly37esolve the singularity in order to apply the localization formula. Notice here thatsince ξ v preserves all the structure on the compact manifold ( Y, g Y , σ ), the closure ofits orbits defines a U (1) s action preserving all the structure, for some s ≥
1. Herewe have used the fact that the isometry group of a compact Riemannian manifold iscompact. Thus (
X, ω ) comes equipped with a U (1) s action.Rather than attempt to describe this in general, we focus here on the special casewhere the solution is toric : that is, there is a U (1) action on Y under which σ , andhence ω under the lift to X , is invariant. Notice that we do not necessarily require thatthe full supergravity solution is invariant under U (1) – we shall illustrate this in thenext section with the Pilch-Warner solution, where σ and the metric are invariant under U (1) , but the G -flux is invariant only under a U (1) subgroup. For the argumentsthat follow, it is only σ (and hence ω ) that we need to be invariant under a maximaldimension torus U (1) . In fact any such symplectic toric cone is also an affine toricvariety. This implies that there is a (compatible) complex structure on X , and thatthe U (1) action complexifies to a holomorphic ( C ∗ ) action with a dense open orbit.There is then always a symplectic toric resolution ( X ′ , ω ′ ) of ( X, ω ), obtained by toricblow-up. In physics language, this is because one can realize (
X, ω ) as a gauged linearsigma model, and one obtains ( X ′ , ω ′ ) by simply turning on generic Fayet-Iliopoulosparameters. One can also describe this in terms of moment maps as follows. Theimage of a symplectic toric cone under the moment map µ : X → R is a strictlyconvex rational polyhedral cone (see [11]). Choosing a toric resolution ( X ′ , ω ′ ) thenamounts to choosing any simplicial resolution P of this polyhedral cone. Here P is theimage of µ ′ : X ′ → R . Assuming the fixed points of ξ v are all isolated, the localizationformula is then simply [11]1(2 π ) Z X e − r / ω
3! = X vertices p ∈P Y i =1 h ξ v , u p i i . (6.25)Here u p i , i = 1 , ,
3, are the three edge vectors of the moment polytope P at the vertexpoint p, and h· , ·i denotes the standard Euclidean inner product on R (where we regard ξ v as being an element of the Lie algebra R of U (1) ). The vertices of P preciselycorrespond to the U (1) fixed points of the symplectic toric resolution X ′ = X P of X .Thus, remarkably, these results of [11] hold in general, even when there are non-trivialfluxes turned on and X is not Calabi-Yau.38 .4 The conformal dimensions of BPS branes A supersymmetric D3-brane wrapped on Σ ⊂ Y gives rise to a BPS particle in AdS .The quantum field Φ whose excitations give rise to this particle state then couples,in the usual way in AdS/CFT, to a dual chiral primary operator O = O Σ in theboundary SCFT. More precisely, there is an asymptotic expansion of Φ near the AdS boundary Φ ∼ Φ r ∆ − + A Φ r − ∆ , (6.26)where Φ acts as the source for O and ∆ = ∆( O ) is the conformal dimension of O .In [39], following [40], it was argued that the vacuum expectation value A Φ of O in agiven asymptotically AdS background may be computed from e − S E , where S E is theon-shell Euclidean action of the D3-brane wrapped on Σ = R + × Σ , where R + is the r -direction. In particular, via the second term in (6.26) this identifies the conformaldimension ∆ = ∆( O Σ ) with the coefficient of the logarithmically divergent part of theon-shell Euclidean action of the D3-brane wrapped on Σ . We refer to section 2.3 of[39] for further details.We are thus interested in the on-shell Euclidean action of a supersymmetric D3-brane wrapped on Σ = R + × Σ . The condition of supersymmetry is equivalent to ageneralized calibration condition, namely equation (3.16) of [35]. In our notation andconventions, this calibration condition readsRe (cid:2) − iΦ + ∧ e F (cid:3) | Σ = | a | p det( h + F ) d x ∧ · · · ∧ d x . (6.27)Here h is the induced (string frame) metric on Σ , and F = F − B is the gauge-invariantworldvolume gauge field, satisfyingd F = − H | Σ . (6.28)Recalling from section 3.2 that | a | = e A , we may then substitute for Φ + in terms ofΩ + using (3.36) and (6.1) to obtainRe (cid:2) − iΦ + ∧ e F (cid:3) | Σ = f
64 e A + φ d log r ∧ σ ∧ d σ | Σ − f
64 e − A + φ r ( F − b + ) | Σ , (6.29)where, as in (6.3), b + = e φ/ f Im K ∧ d log r + b . (6.30)39ere b is a closed two-form, whose gauge-invariant information is contained in itscohomology class in H ( X, R ) /H ( X, Z ). In writing b + in (6.29) we have chosen a par-ticular representative two-form for the class of b in H ( X, R ) /H ( X, Z ). Then underany gauge transformation of b + (induced from a B -transform of Ω + ), the worldvolumegauge field F transforms by precisely the opposite gauge transformation restricted toΣ , so that the quantity F − b + is gauge invariant on Σ . We now choose the world-volume gauge field F to be F = b | Σ , (6.31)so that (6.29) becomes simplyRe (cid:2) − iΦ + ∧ e F (cid:3) | Σ = f
64 e A + φ d log r ∧ σ ∧ d σ | Σ . (6.32)In fact, there is a slight subtlety in (6.31). If the cohomology class of b / (2 πl s ) | Σ in H (Σ , R ) is not integral, then the choice (6.31) is not possible as F is the curvatureof a unitary line bundle. Having said this, notice H (Σ , R ) ∼ = H (Σ , R ), and thusin particular that if H (Σ , R ) = 0 then every closed b | Σ is exact, and thus may begauge transformed to zero on Σ . Then (6.31) simply sets F = 0. For every exampleof a supersymmetric Σ that we are aware of, this is indeed the case. In any case, weshall assume henceforth that the choice (6.31) is possible.The calibration condition (6.27) for a D3-brane with worldvolume Σ and with gaugefield (6.31) is thus f r ∧ σ ∧ d σ = e − φ p det( h − B ) d x ∧ · · · ∧ d x . (6.33)Notice the right hand side is precisely the Dirac-Born-Infeld Lagrangian, up to the D3-brane tension τ = 1 / (2 π ) l s g s . From (6.33), and the comments above on the scalingdimension ∆( O (Σ )) of the dual operator O (Σ ), we thus deduce∆( O (Σ )) = − τ f Z Σ σ ∧ d σ . (6.34)(The sign just arising from a convenient choice of orientation.) Using (6.19) and (6.5)we have f = − πl s ) g s N R Y σ ∧ d σ ∧ d σ , (6.35)and hence ∆( O (Σ )) = 2 πN R Σ σ ∧ d σ R Y σ ∧ d σ ∧ d σ . (6.36)40his is our final formula for the conformal dimension of the chiral primary operatordual to a BPS D3-brane wrapped on Σ . Since we may write Z Σ σ ∧ d σ = Z Σ e − r / ω , (6.37)we again see that it depends only on the symplectic structure of ( X, ω ) and the Reebvector field ξ v . This again may be evaluated by localization, having appropriatelyresolved the tip of the cone Σ . In this section we illustrate the general results derived so far with the Pilch-Warnersolution [13]. (Some aspects of the generalized complex geometry of this backgroundhave already been discussed in [20].) Recall that the Pilch-Warner solution is dualto a Leigh-Strassler fixed point theory [16] which is obtained by giving a mass to oneof the three chiral superfields (in N = 1 language) of N = 4 SU ( N ) super-Yang-Mills theory, and following the resulting renormalization group flow to the IR fixedpoint theory. This latter theory is an N = 1 SU ( N ) gauge theory with two adjointfields Z a , a = 1 ,
2, which form a doublet under an SU (2) flavour symmetry, and aquartic superpotential. Since the superpotential has scaling dimension three, this fixes∆( Z a ) = 3 /
4, implying that the IR theory is strongly coupled. The mesonic modulispace is simply Sym N C .The Pilch-Warner supergravity solution [13] was rederived in [19], and we shall usesome of the results from that reference also. We have Y = S with non-trivial metric g Y = 19 (cid:20) ϑ + 6 cos ϑ − cos 2 ϑ ( σ + σ ) + 6 sin ϑ (3 − cos 2 ϑ ) σ +4 (cid:18) d ϕ + 2 cos ϑ − cos 2 ϑ σ (cid:19) (cid:21) , (7.1)where 0 ≤ ϑ ≤ π , 0 ≤ ϕ ≤ π , and σ i , i = 1 , ,
3, are left-invariant one-forms on SU (2)(denoted with hats in [19]). The dilaton φ and axion C are simply constant, while thewarp factor is e = f − cos 2 ϑ ) . (7.2)41here is also a non-trivial NS and RR three-form flux given by (see (A.7)) G = (2 f ) / / e ϕ cos ϑ d ϕ ∧ d ϑ − i sin 2 ϑ − cos 2 ϑ d ϕ ∧ σ − ϑ (3 − cos 2 ϑ ) d ϑ ∧ σ ! ∧ ( σ − i σ ) . (7.3)We introduce the Euler angles ( α, β, γ ) on SU (2) (as in [19]), so that σ = − sin γ d α − cos γ sin α d β ,σ = cos γ d α − sin γ sin α d β ,σ = d γ − cos α d β . (7.4)In terms of these coordinates, the R -symmetry vector ξ v is [19] ξ v = 32 ∂ ϕ − ∂ γ . (7.5)Using the explicit formulae in [19], it is easy to show that the contact form is σ = − (cid:0) cos 2 ϑ d ϕ + cos ϑ σ (cid:1) . (7.6)The solution is toric, in the sense that both σ and the metric are invariant undershifts of ϕ , β and γ . However, notice that the G -flux in (7.3) is not invariant undershifts of ϕ , thus breaking this U (1) symmetry to only a U (1) symmetry of the fullsupergravity solution. This is expected, since the dual field theory described above hasonly an SU (2) × U (1) R global symmetry.On Y = S there are precisely three invariant circles under the U (1) action, wheretwo of the U (1) actions degenerate, namely at { ϑ = π } , { ϑ = 0 , α = 0 } , { ϑ = 0 , α = π } .A set of 2 π -period coordinates on U (1) are ϕ = ϕ, ϕ = −
12 ( ϕ + γ − β ) , ϕ = −
12 ( ϕ + γ + β ) . (7.7)These restrict to coordinates on the above three invariant circles, respectively. On X ∼ = R \ µ = r ϑ, µ = r ϑ (1 + cos α ) , µ = r ϑ (1 − cos α ) , (7.8)so that ω = d( r σ ) = P i =1 d µ i ∧ d ϕ i . It follows that the image of the moment map –the space spanned by the µ i coordinates – is the cone ( R ≥ ) , where the three invariant42ircles map to the three generating rays u = (1 , , u = (0 , , u = (0 , , ξ = 32 ∂ ϕ − ∂ γ = 32 ∂ ϕ + 34 ∂ ϕ + 34 ∂ ϕ . (7.9)Since the symplectic structure is smooth at r = 0, we may evaluate (6.25) by local-ization without having to resolve X at r = 0. In the case at hand, we have the singlefixed point at r = 0, and from (7.9) one obtains the known result a N =4 a P W = 1 ξ ξ ξ = 3227 . (7.10)They key point about the above calculation is that we have computed this knowingonly the symplectic structure of the solution and the Reeb vector field ξ v .We may similarly compute the conformal dimensions of the operators det Z a , using(6.36), by interpreting them as arising from a BPS D3-brane wrapped on the three-spheres at α = 0 and α = π , respectively. It is simple to check these indeed satisfy thecalibration condition (6.33) and are thus supersymmetric. Using (6.37) and localizationat r = 0 implies that (6.37) is equal to 1 /ξ ξ , 1 /ξ ξ , respectively, which in both casesis 8 /
9. The formula (6.36) thus gives ∆(det Z a ) = 3 N/
4, or equivalently ∆( Z a ) = 3 / θ = d( r θ ), where θ = − e S , and themesonic moduli space should be the locus θ = 0. As discussed in section 5.2, this isthe locus sin 2¯ θ = sin 2 ¯ φ = 0. For the Pilch-Warner solution, we may easily computesin 2¯ θ = − √ ϑ √ ϑ , cos 2 ¯ φ = √ ϑ ϑ . (7.11)Thus, as discussed in [20], the mesonic moduli space S = 0 is equivalent to ϑ = 0,which is a codimension two submanifold in R diffeomorphic to R . Moreover, this is C in the induced complex structure, and we thus see explicit agreement with the fieldtheory N = 1 mesonic moduli space.Finally, although the Pilch-Warner solution is generalized complex, rather thancomplex, we note that one can nevertheless define a natural complex structure [41].The relation between this integrable complex structure and the generalized geometryhas been discussed in [20]. Let us conclude this section by elucidating this connection.One can introduce the following complex coordinates [20] in terms of the angular43ariables (7.7): s = r / sin ϑ e − i ϕ ,s = r / cos ϑ cos α e i ϕ ,s = r / cos ϑ sin α e i ϕ . (7.12)This makes R ∼ = C . However, because of the minus sign in the first coordinate in(7.12), the corresponding integrable complex structure I ∗ is not the unique complexstructure that is compatible with the toric structure of the solution: the latter insteadhas complex coordinates ¯ s , s , s . Indeed, also the Reeb vector field ξ v is not given by I ∗ ( r∂ r ). This makes the physical significance of this complex structure rather unclear.Nevertheless, one can show that I ∗ does in fact come from an SU (3) structure definedby a Killing spinor. Following [20], we define2ˆ aη ∗ = η + i η = e A/ ξ i ξ ! , (7.13)where by definition we require ¯ η ∗ η ∗ = 1. It is then convenient to define ˆ a ≡ | ˆ a | e i z ,where | ˆ a | = e A | ξ | = e A (1 − sin ζ ). We then introduce the bilinears correspondingto the SU (3) structure defined by η ∗ : J ∗ ≡ − i¯ η ∗ γ (2) η ∗ , (7.14)Ω ∗ ≡ ¯ η c ∗ γ (3) η ∗ . (7.15)One computes that dΩ ∗ = 0, implying that the corresponding complex structure I ∗ isintegrable, and moreover thate z Ω ∗ = − e α √ f / A d s ∧ d s ∧ d s , (7.16)implying that (7.12) are indeed complex coordinates for this complex structure. Wealso compute J ∗ = − e A r h d log r ∧ (cid:18) d ϕ + cos ϑ (1 + sin ϑ ) σ (cid:19) + 13(1 + sin ϑ ) (cid:0) sin 2 ϑσ ∧ d ϑ + cos ϑσ ∧ σ (cid:1)i . (7.17) In this paper we have initiated an analysis of the generalized cone geometry associ-ated with supersymmetric
AdS × Y solutions of type IIB supergravity. The cone is44eneralized Hermitian and generalized Calabi-Yau and we have identified holomorphicgeneralized vector fields that are dual to the dilatation and R -symmetry of the dualSCFT. We identified a relationship between “BPS polyforms”, i.e. polyforms withequal R -charge and scaling weight, and generalized holomorphic polyforms that shouldbe worth exploring further. In particular, we would like to make a precise connec-tion between such objects and the the spectrum of chiral operators in the SCFT viaKaluza-Klein reduction on Y .We also showed how one can carry out a generalized reduction of the six-dimensionalcone to obtain a new four-dimensional transverse generalized Hermitian geometry. Thisgeneralizes the transverse K¨ahler-Einstein geometry in the Sasaki-Einstein case. Byanalogy with the Sasaki-Einstein case ( e.g. [25]) this perspective could be useful forconstructing new explicit solutions.We also analysed the symplectic structure on the cone geometry, which exists pro-viding that the five-form flux is non-vanishing. It would be interesting to know whetheror not this includes all solutions. We obtained Duistermaat-Heckman type integralsfor the central charge of the dual SCFT and the conformal dimensions of operatorsdual to BPS wrapped D3-branes. These formulae precisely generalize analogous for-mulae that were derived in [10, 11] for the Sasaki-Einstein case. Other formulae forthese quantities were also presented in [10, 11] and we expect that these will also haveprecise generalizations in terms of generalized geometry. In particular, we expect ageneralized geometric interpretation of a -maximization. Acknowledgments
We would like to thank Nick Halmagyi for a useful discussion. M.G. is supported bythe Berrow Foundation, J.P.G. by an EPSRC Senior Fellowship and a Royal SocietyWolfson Award, E.P. by a STFC Postdoctoral Fellowship and J.F.S. by a Royal SocietyUniversity Research Fellowship.
A Conventions and 6D to 5D map
We use exactly the same conventions as in [19], up to some simple relabelling. Herewe will explain how the results of that paper concerning the five-dimensional geometry Although it will not be relevant in this paper we point out that there is a typo in [19]: the ρ a matrices generating Cliff(4 ,
1) actually satisfy ρ = − i. g Y can be uplifted to six-dimensions. In particular, we will relate the five-dimensional Killing spinors discussed in [19] to the six-dimensional chiral spinors η i thatdefine the bispinors Φ ± . We first recall the Killing spinor equations in five-dimensions,related to the geometry g Y , given in [19]. There are two differential conditions( ∇ m − i2 Q m ) ξ + i4 (cid:0) e − f − (cid:1) β m ξ + 18 e − G mnp β np ξ = 0 , (A.1)( ∇ m + i2 Q m ) ξ − i4 (cid:0) e − f + 2 (cid:1) β m ξ + 18 e − G ∗ mnp β np ξ = 0 , (A.2)and four algebraic conditions β m ∂ m ∆ ξ −
148 e − β mnp G mnp ξ − i4 (cid:0) e − f − (cid:1) ξ = 0 , (A.3) β m ∂ m ∆ ξ −
148 e − β mnp G ∗ mnp ξ + i4 (cid:0) e − f + 4 (cid:1) ξ = 0 , (A.4) β m P m ξ + 124 e − β mnp G mnp ξ = 0 , (A.5) β m P ∗ m ξ + 124 e − β mnp G ∗ mnp ξ = 0 . (A.6)Here the β m generate the Clifford algebra for g Y , so { β m , β n } = 2 g Y mn . Equivalently,with respect to any orthonormal frame the corresponding ˆ β m satisfy { ˆ β m , ˆ β n } = 2 δ mn .We have chosen ˆ β = +1. In addition we have set the parameter m in [19] to be m = 1, consistent with (3.2). In the usual string theory variables we have P = 12 d φ + i2 e φ F ,Q = −
12 e φ F ,G = − ie φ/ F − e − φ/ H , (A.7)where the RR field strengths F n are defined by (3.3). We also note that the constant f appearing in the Killing spinor equations is related to the component of the self-dualfive-form flux on Y (3.6) via F | Y = − f f vol Y , (A.8)where the five-dimensional volume form is defined as f vol Y = − e and e i is theorthonormal frame introduced in appendix B of [19].We now provide a map between the five-dimensional spinors and Killing spinorequations (A.1)-(A.6) to six-dimensional quantities. We begin by using the Cliff(5) Notice we have relabelled γ i β m in [19], as in this paper we want to keep the notation γ i forsix-dimensional gamma matrices. β m to construct Cliff(6) gamma matrices ˆ γ i , i = 1 , . . . ,
6, viaˆ γ m = ˆ β m ⊗ σ , m = 1 , . . . , γ = 1 ⊗ σ , (A.9)where σ α , α = 1 , ,
3, are the Pauli matrices. These satisfy { ˆ γ i , ˆ γ j } = 2 δ ij . Thecorresponding gamma matrices for the six-dimensional metric g will be denoted γ i .We define the 6D chirality operator to be˜ γ ≡ − iˆ γ = 1 ⊗ σ . (A.10)We may choose the D intertwiner D = ˜ D ⊗ σ , (A.11)where ˜ D = D = C is the intertwiner of Cliff(5) discussed in [19], and one checks D − γ i D = − γ ∗ i . We also note that since in [19] the interwiner A = 1 we have A = 1and γ † i = γ i . If η + is a Weyl spinor, satisfying ˜ γη + = η + , then the conjugate spinor η − ≡ η c + ≡ D η ∗ + satisfies ˜ γη − = − η − .To construct the relevant 6D spinors we first write ξ = χ + i χ , ξ = χ − i χ , (A.12)as in [19]. Given this, the normalization for ξ i chosen in [19] implies that the χ i arenormalized as ¯ χ χ = ¯ χ χ = . (A.13)We then define η = e A/ χ i χ ! , η − = e A/ − χ c i χ c ! ,η = e A/ − χ − i χ ! , η − = e A/ χ c − i χ c ! , (A.14)where recall from (3.16) that e A/ = r / e ∆ / φ/ , (A.15)and also from [19] that χ c ≡ ˜ D χ ∗ . (A.16)47n the conventions of [19] we have ¯ χ = χ † .After some detailed calculation one finds that the five-dimensional Killing spinorequations (A.1)–(A.6), using the five-dimensional metric g Y , are equivalent to the six-dimensional Killing spinor equations, using the six-dimensional metric g in (3.17) andvolume form (3.18), given by (cid:18) D i − H i (cid:19) η + e φ F γ i η = 0 , (A.17)12 e A ∂A η −
18 e A + φ F η = 0 , (A.18) Dη + (cid:16) ∂ (2 A − φ ) − H (cid:17) η = 0 , (A.19)and additional equations obtained by applying the rule: η ↔ η , F → −6 F † , H → − H . (A.20)In these equations we are using the notation that, e.g. H i = H ijk γ jk , F = F i γ i + F ijk γ ijk + F ijklm γ ijklm . (A.21)These are precisely the same equations that were used in [34] (for zero four-dimensionalcosmological constant).Finally, we record the following equation of [19]: D (e W ) = − e P ∧ W ∗ + f G , (A.22)where W is the two-form bilinear defined in (3.10). Using this one can show that i K (cid:18) f e φ/ Re W (cid:19) = e φ Re K , (A.23)and furthermore that d (cid:18) e φ Re K (cid:19) = i ξ v H . (A.24)To see the latter one can derive an expression for the left hand side using, amongstother things, (3.18), (3.38) and (B.10) of [19], and an expression for the right hand sideusing equation (3.38) and (B.8) of [19]. Using these results we can deduce that L K B = d( i K b ) , L K C = d( i K c ) , (A.25)where b , c were introduced in (3.12), (3.13), respectively.48 More on the generalized vectors ξ and η In this appendix we derive an expression for the generalized vector ξ in terms of thebilinears introduced in [19]. We also use the results of [19] to show that L ξ J ± = 0.The projections of ξ onto the vector and form parts (in a fixed trivialization of E )are denoted ξ v , ξ f , respectively. It will also be convenient to introduce ξ B ≡ e B ξ whoseform part is given by ξ Bf = ξ f − i ξ v B . (B.1)and we recall that ξ Bv = ξ v . We next construct the following two generalized (1 , − vectors, which, by definition, are in the +i eigenspace of J − : Z − = r∂ r − i ξ ,Z − = d log r − i η . (B.2)That is, both are in the annihilator of Ω − . We may similarly also construct the (1 , + vectors, with respect to J + : Z +1 = e − ∆ − φ/ r∂ r − ie ∆+ φ/ η ,Z +2 = e ∆+ φ/ d log r − ie − ∆ − φ/ ξ . (B.3)Together Z ± i are four independent generalized vectors. We next note that since Z ± i livewithin null isotropic subspaces we have six relations of the form, using the notation of(2.5), (cid:10) Z ± i , Z ± j (cid:11) = 0 . (B.4)Explicitly we have i ξ v ξ f = 0 , i ξ v d log r = 0 , i r∂ r ξ f = 0 ,i η v η f = 0 , i η v d log r = 0 , i r∂ r η f = 0 , h ξ, η i = . (B.5)Since Z − annihilates Ω − , using the definition (3.34) we deduce that i r∂ r Φ − = i (cid:0) i ξ v Φ − + ξ Bf ∧ Φ − (cid:1) . (B.6)To proceed we use (3.20) to writeΦ − ≡ η ⊗ ¯ η − = e A χ ¯ χ c ⊗ ( σ + i σ ) . (B.7)Since Φ − = P odd n n ! Φ i ...i n γ i ...i n we have i v Φ − = { v i γ i , Φ − } , ω ∧ Φ − = [ ω i γ i , Φ − ] . (B.8)49ence, using the Clifford algebra decomposition (3.19) and metric (3.17) we have i r∂ r Φ − = { e ∆+ φ/ ˆ γ , Φ − } = e A +∆+ φ/ χ ¯ χ c ⊗ { σ , σ + i σ } = ie A +∆+ φ/ χ ¯ χ c ⊗ . (B.9)On the other hand using (B.5) we have i ξ v Φ − + ξ Bf ∧ Φ − = { e ∆+ φ/ ξ mv β m ⊗ σ , Φ − } + [e − ∆ − φ/ ξ Bfm β m ⊗ σ , Φ − ]= e ∆+ φ/ v + m β m ⊗ σ Φ − + e ∆+ φ/ v − m Φ − β m ⊗ σ = e A +∆+ φ/ (cid:0) v + m β m ( χ ¯ χ c ) + v − m ( χ ¯ χ c ) β m (cid:1) ⊗ − e A +∆+ φ/ (cid:0) v + m β m ( χ ¯ χ c ) − v − m ( χ ¯ χ c ) β m (cid:1) ⊗ σ , (B.10)where recall that { β m , β n } = 2 g Y mn and we have defined v ± m = ( ξ vm ± e − − φ/ ξ Bfm ) . (B.11)To satisfy (B.6) we thus require v + m β m ( χ ¯ χ c ) = v − m ( χ ¯ χ c ) β m = χ ¯ χ c , (B.12)which implies v + m β m χ = χ , v − m β m χ = χ , (B.13)or equivalently v + m = ¯ χ β m χ χ χ , v − m = ¯ χ β m χ χ χ . (B.14)Hence, given the normalizations (A.13) we deduce that, in terms of the bilinears definedin (3.9), ξ v = K ,ξ Bf = e φ/ Re K . (B.15)A similar calculation using i r∂ r Φ + = ie φ/ (cid:0) i η v Φ − + η Bf ∧ Φ + (cid:1) , (B.16)leads to η v = e − − φ/ Re K ,η Bf = K . (B.17)Using the expression for the B -field given in (3.12) we obtain the expressions for ξ f and η f given in (3.50). 50n [19] it was shown that K is a Killing one-form, so that its dual vector field K ,with respect to the metric g Y on Y , is a Killing vector field. In fact K generates a fullsymmetry of the supergravity solution, in that all bosonic fields (warp factor, dilaton,NS three-form H and RR fields) are preserved under the Lie derivative along ξ v = K .However, importantly, the Killing spinors ξ , ξ are not invariant under ξ v . In [19] itwas shown that L ξ v S = − S , (B.18)where S ≡ ¯ ξ c ξ . Notice that, since ξ v preserves all of the bosonic fields, one may takethe Lie derivative of the Killing spinor equations (A.1)-(A.6) for ξ , ξ along ξ v , showingthat {L ξ v ξ i } satisfy the same equations as the { ξ i } . It thus follows that L ξ v ξ i = i µξ i , (B.19)where µ is a constant. Now (B.18) implies that 2 µ = −
3, and thus L ξ v ξ i = − ξ i . (B.20)One can also derive this last equation directly from the Killing spinor equations (A.1)-(A.6) of [19]. It thus follows that L ξ v Φ + = 0 , (B.21) L ξ v Φ − = −
3i Φ − . (B.22)From (A.24) we have d ξ Bf = i ξ v H and we deduce that L ξ B Φ + = i ξ v H ∧ Φ + , L ξ B Φ − = −
3i Φ − + i ξ v H ∧ Φ − . (B.23)Since (A.24) is also equivalent to d ξ f = L ξ v B we deduce that L ξ Ω + = 0 , L ξ Ω − = − − , (B.24)and hence L ξ J ± = 0. It is also interesting to point out that( L ξ B − i ξ v H ∧ ) F = 0 , or equivalently , L ξ (e − B F ) = 0 . (B.25)51 The Sasaki-Einstein case
Here we discuss the special case in which the compact five-manifold Y is Sasaki-Einstein. Setting G = P = Q = 0, f = 4 e and ξ = 0, the Killing spinor equations(A.1)-(A.6) reduce to ∇ m ξ + i2 β m ξ = 0 . (C.1)In terms of appendix B of [19] we choose ¯ θ = ¯ φ = 0 and e α = − η = 12 ¯ ξ β (1) ξ = K = e ,ω KE = i2 ¯ ξ β (2) ξ = − V = e + e , Ω KE = 12 ¯ ξ β (2) ξ c = ( e + i e ) ∧ ( e + i e ) , (C.2)and d η = 2 ω KE , dΩ KE = 3i η ∧ Ω KE . (C.3)Observe that η ∧ ω = − e = f vol Y . (C.4)Next using the 5D-6D map (3.20), we obtaini¯ η γ (2) η = r (d log r ∧ e + ω KE ) ≡ r ω CY , − i¯ η c + γ (3) η = r (d log r − i e )( e − i e )( e − i e ) ≡ r ¯Ω CY . (C.5)It is worth noting that13! ω = r e = r d log r ∧ η ∧ ω . (C.6)We also find, directly from (3.33), (3.36)Ω − = 18 ¯Ω CY , Ω + = − i r (cid:18) i r ω CY (cid:19) . (C.7)A useful check is that these expressions agree with those obtained from the generalexpressions obtained in sections 5.1 and 6.1, respectively.52ne can also write down the corresponding reduced structures ˜Ω and ˜Ω , as definedin section 4.3, on the K¨ahler-Einstein space. One finds˜Ω = 18 e ψ ¯Ω KE , ˜Ω = − i8 e i ω KE . (C.8)where ψ is the coordinate, defined such that K = ∂ ψ , introduced in (4.31). References [1] J. P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, “Supersymmetric
AdS so-lutions of M-theory,” Class. Quant. Grav. (2004) 4335 [arXiv:hep-th/0402153].[2] J. P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, “Sasaki-Einstein metricson S × S ,” Adv. Theor. Math. Phys. (2004) 711 [arXiv:hep-th/0403002].[3] M. Cvetic, H. Lu, D. N. Page and C. N. Pope, “New Einstein-Sasakispaces in five and higher dimensions,” Phys. Rev. Lett. (2005) 071101[arXiv:hep-th/0504225].[4] S. Benvenuti, S. Franco, A. Hanany, D. Martelli and J. Sparks, “An infinite familyof superconformal quiver gauge theories with Sasaki-Einstein duals,” JHEP (2005) 064 [arXiv:hep-th/0411264].[5] A. Hanany and K. D. Kennaway, “Dimer models and toric diagrams,”arXiv:hep-th/0503149.[6] S. Franco, A. Hanany, K. D. Kennaway, D. Vegh and B. Wecht, “Brane Dimersand Quiver Gauge Theories,” JHEP (2006) 096 [arXiv:hep-th/0504110].[7] S. Franco, A. Hanany, D. Martelli, J. Sparks, D. Vegh and B. Wecht, “Gaugetheories from toric geometry and brane tilings,” JHEP (2006) 128[arXiv:hep-th/0505211].[8] D. Anselmi, J. Erlich, D. Z. Freedman and A. A. Johansen, “Positivity constraintson anomalies in supersymmetric gauge theories,” Phys. Rev. D , 7570 (1998)[arXiv:hep-th/9711035].[9] K. A. Intriligator and B. Wecht, “The exact superconformal R-symmetry maxi-mizes a ,” Nucl. Phys. B , 183 (2003), [arXiv:hep-th/0304128].5310] D. Martelli, J. Sparks and S. T. Yau, “The geometric dual of a -maximisationfor toric Sasaki-Einstein manifolds,” Commun. Math. Phys. , 39 (2006)[arXiv:hep-th/0503183].[11] D. Martelli, J. Sparks and S. T. Yau, “Sasaki-Einstein manifolds and volumeminimisation,” Commun. Math. Phys. , 611 (2008) [arXiv:hep-th/0603021].[12] O. Lunin and J. M. Maldacena, “Deforming field theories with U (1) × U (1) global symmetry and their gravity duals,” JHEP , 033 (2005)[arXiv:hep-th/0502086].[13] K. Pilch and N. P. Warner, “A new supersymmetric compactification of chiral IIBsupergravity,” Phys. Lett. B , 22 (2000) [arXiv:hep-th/0002192].[14] A. Khavaev, K. Pilch and N. P. Warner, “New vacua of gauged N = 8 supergravityin five dimensions,” Phys. Lett. B (2000) 14 [arXiv:hep-th/9812035].[15] N. Halmagyi, K. Pilch, C. Romelsberger and N. P. Warner, “Holographic duals ofa family of N = 1 fixed points,” JHEP , 083 (2006) [arXiv:hep-th/0506206].[16] R. G. Leigh and M. J. Strassler, “Exactly Marginal Operators And Duality InFour-Dimensional N = 1 Supersymmetric Gauge Theory,” Nucl. Phys. B , 95(1995) [arXiv:hep-th/9503121].[17] S. Benvenuti and A. Hanany, “Conformal manifolds for the conifold and othertoric field theories,” JHEP (2005) 024 [arXiv:hep-th/0502043].[18] O. Aharony, B. Kol and S. Yankielowicz, “On exactly marginal deformations of N = 4 SYM and type IIB supergravity on AdS × S ,” JHEP (2002) 039[arXiv:hep-th/0205090].[19] J. P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, “Supersymmetric AdS solutions of type IIB supergravity,” Class. Quant. Grav. , 4693 (2006)[arXiv:hep-th/0510125].[20] R. Minasian, M. Petrini and A. Zaffaroni, “Gravity duals to deformedSYM theories and generalized complex geometry,” JHEP , 055 (2006)[arXiv:hep-th/0606257]. 5421] A. Butti, D. Forcella, L. Martucci, R. Minasian, M. Petrini and A. Zaffaroni,“On the geometry and the moduli space of beta-deformed quiver gauge theories,”JHEP , 053 (2008) [arXiv:0712.1215 [hep-th]].[22] M. Grana, R. Minasian, M. Petrini and A. Tomasiello, “Supersymmetric back-grounds from generalized Calabi-Yau manifolds,” JHEP , 046 (2004) [arXiv:hep-th/0406137].[23] M. Grana, R. Minasian, M. Petrini and A. Tomasiello, “Generalized structures of N = 1 vacua,” JHEP , 020 (2005) [arXiv:hep-th/0505212].[24] N. Hitchin, “Generalized Calabi-Yau manifolds,” Quart. J. Math. Oxford Ser. (2003) 281-308 [arXiv:math/0209099].[25] J. P. Gauntlett, D. Martelli, J. F. Sparks and D. Waldram, “A new infiniteclass of Sasaki-Einstein manifolds,” Adv. Theor. Math. Phys. (2006) 987[arXiv:hep-th/0403038].[26] H. Bursztyn, G. Cavalcanti and M. Gualtieri “Reduction of Courant alge-broids and generalized complex structures”, Adv. Math. , 726 (2007) [arXiv:math/0509640v3 [math.DG]].[27] H. Bursztyn, G. Cavalcanti and M. Gualtieri “Generalized Kahler and hyper-Kahler quotients”, arXiv:math/0702104v1 [math.DG].[28] M. Gabella, J. P. Gauntlett, E. Palti, J. Sparks and D. Waldram, “The centralcharge of supersymmetric AdS solutions of type IIB supergravity,” Phys. Rev.Lett. (2009) 051601 [arXiv:0906.3686 [hep-th]].[29] M. Gualtieri, “Generalized complex geometry,” arXiv:math/0703298.[30] N. Hitchin, “Brackets, forms and invariant functionals,” arXiv:math/0508618.[31] M. Grana, R. Minasian, M. Petrini and D. Waldram, “T-duality, GeneralizedGeometry and Non-Geometric Backgrounds,” arXiv:0807.4527 [hep-th].[32] G. R. Cavalcanti, “New aspects of the dd c -lemma,” arXiv:math.dg/0501406.[33] M. Gualtieri, “Generalized complex geometry,” Oxford University DPhil thesis,arXiv:math/0401221 [math-DG]. 5534] M. Grana, R. Minasian, M. Petrini and A. Tomasiello, “A scan for new N = 1vacua on twisted tori,” JHEP (2007) 031 [arXiv:hep-th/0609124].[35] L. Martucci and P. Smyth, “Supersymmetric D-branes and calibrations on generalN = 1 backgrounds,” JHEP , 048 (2005) [arXiv:hep-th/0507099].[36] A. Tomasiello, “Reformulating Supersymmetry with a Generalized Dolbeault Op-erator,” JHEP , 010 (2008) [arXiv:0704.2613 [hep-th]].[37] L. Martucci, “D-branes on general N = 1 backgrounds: Superpotentials and D-terms,” JHEP , 033 (2006) [arXiv:hep-th/0602129].[38] M. Henningson and K. Skenderis, “The holographic Weyl anomaly,” JHEP ,023 (1998) [arXiv:hep-th/9806087].[39] D. Martelli and J. Sparks, “Baryonic branches and resolutions of Ricci-flat Kahlercones,” JHEP , 067 (2008) [arXiv:0709.2894 [hep-th]].[40] I. R. Klebanov and A. Murugan, “Gauge/Gravity Duality and Warped ResolvedConifold,” JHEP , 042 (2007) [arXiv:hep-th/0701064].[41] N. Halmagyi, K. Pilch, C. Romelsberger and N. P. Warner, “The complex ge-ometry of holographic flows of quiver gauge theories,” JHEP0609