Marginal deformations of a class of AdS_3 \mathcal{N}=(0,4) holographic backgrounds
PPrepared for submission to JHEP
Marginal deformations of a class of AdS N = (0 , holographic backgrounds Salomon Zacarías a a Department of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University611 37 Brno, Czech Republic
E-mail: [email protected]
Abstract:
We discuss marginal deformations of warped AdS × S solutions preservingsmall N = (0 , supersymmetry in massive IIA and eleven-dimensional supergravity andobtain a whole family of new solutions. We characterise these new backgrounds by studyingsome observables like the quantised charges, associated Hannany-Witten brane set-ups andthe holographic central charge, the latter is shown to be invariant under the deformation.The study of the preservation of supersymmetry shows that the new backgrounds supportan identity structure on the internal five-dimensional space, which is dynamical. a r X i v : . [ h e p - t h ] F e b ontents N = (0 , holographic backgrounds 33 The marginally deformed backgrounds 4 Supersymmetric solutions with AdS p +1 factors in type II and eleven-dimensional super-gravities play a prominent role in the context of the AdS/CFT correspondence since theyprovide a holographic description of p-dimensional superconformal field theories (SCFTs)at strong coupling [1]. The discovery of this holographic duality has ever since triggered anumber of efforts to construct and classify AdS vacua for any dimension allowed and pre-serving various amounts of (super)symmetries that have been used to study and characteriseSCFTs.More recently, the case of AdS backgrounds has gained a lot of attention. There areseveral motivations for this. For instance, the near horizon geometries of five-dimensionalextremal black holes have AdS factors. Using SCFT data it is then possible to understandmicroscopic features of black holes, like their entropy by computing the central charge ofthe SCFT [2], among other aspects (see for instance [3–7]). On the other hand, two-dimensional SCFTs are special on their own since they can, in certain cases, be fully solvabledue to the structure of the superconformal algebra. It is therefore interesting to exploredeeply each side of this dual pair in order to shed some light on new phenomena viaholography. For a sample of works regarding AdS supersymmetric backgrounds in ten andeleven-dimensional supergravity preserving different amounts of supersymmetry and theirholographic applications see [8–41].Moreover, on the geometrical side, attempts to constructing and classifying supersym-metric AdS solutions have been mostly focused on the G-structure formalism [42] for which– 1 –ne extracts geometric constraints for the fields of the solutions according to the numberof (super)symmetries and geometrical structures, etc, we impose in the internal space. Ithas also been considered back-reacting D-brane arrangements which are known to produceAdS solutions in the near horizon limit [34, 36], among others. However, these efforts havebeen non-exhaustive due to the many choices we have on the number of supersymmetries,and superconformal algebras, supported by the solutions constraining the internal spacesubmanifolds. Thus the approach has been focussed on searching and classifying all super-gravity solutions preserving given amounts of supersymmetry, choices of internal structures,etc. This program has allowed to expand significatively our knowledge of new string back-grounds which may have very interesting applications in the context of holography. Inthis vein, another possibility to explore the landscape of AdS vacua is to consider AdS-preserving deformations of well-known supergravity solutions. Depending on the details ofthe deformation these solutions may preserve supersymmetry whilst changing the struc-ture of the internal space, and in some cases escape from presently known classifications ofsupergravity solutions.In this work we will use TsT transformations [43] and the analog to eleven dimensions[44] in order to generate a larger class of warped AdS supersymmetric solutions. The seedbackgrounds we will consider are a subclass of the solutions constructed in [22] which aresolutions of massive IIA supergravity of the warped form AdS × S × CY foliated over aninterval, the two-sphere realising geometrically the SU(2) R R-charge of the solution. Theyare given in terms of three linear functions, preserve small N = (0 , supersymmetry andan SU(2) structure in the internal five-dimensional space, analogously, for these solutions,an SU(3) structure in the seven-dimensional space transverse to AdS . The above solutionsappear in the near horizon limit of D -D -NS5-D -D brane arrangements. D and D branes are colour branes and are suspended between the NS5 branes whilst D and D correspond to localised sources and provide flavour groups attached to the gauge nodeswhich leave the dual quiver CFT anomaly free [23, 24]. For vanishing Romans mass, theuplift to eleven dimensions of the above solution gives rise to a class of AdS × S / Z k × CY foliated over an interval, which preserve the same amount of supersymmetry and internalstructure group [31]. The brane configuration for this solution involves M2 branes and KKmonopoles suspended between M5’ branes as well as extra flavour M5 branes.Given the internal symmetries of the seed solutions above, we have two choices whichproduce inequivalent backgrounds after TsT transformations. Namely, if we consider or notthe azimutal direction inside the S for the process. In the latter case we are left with solu-tions for which supersymmetry is fully preserved. Of course more generic supersymmetricsolutions can be generated by a sequence of TsT’s not involving the U(1) inside the S , butwe will explore this more generic case in the future. For the ten-dimensional solutions, westudy the brane configurations that we propose generate our solutions in the near-horizonlimit. Holographically, these new backgrounds are dual to marginal deformations of theseed (undeformed) SCFT, both theories having the same central charge in the holographiclimit. The latter can be understood since their degrees of freedom, in the aforementionedlimit, are associated to the weighted volume of the internal spaces (to be defined below).The deformation changed the internal space enriching the geometric structure but left in-– 2 –ariant its weighted volume. We prove this using the holographic calculation and left thespecification of the SCFT for a forthcoming publication.The content of this paper is organised as follows. In Section 2 we start by brieflyreviewing the seed solutions in [22]. We then proceed to apply the TsT transformation inSection 3 in order to obtain the new family of backgrounds in massive IIA. We study thequantised charges and present brane configurations which we argue give rise to our solutionsin the near horizon limit. In Section 4 we study the eleven-dimensional analog of the TsTtransformation for the solutions in Section 2 with vanishing romans mass uplifted to elevendimensions. One of the solutions obtained correspond to the uplift of the TsT-deformed IIAsolution in the massless case. We then prove the invariance of the central charge under thedeformation in Section 5 using the holographic computation. Finally, In Section 6 we studythe preservation of supersymmetry for the solutions obtained in Sections 3 and 4 . Thisanalysis suggest the new supersymmetric solutions support a dynamical identity structurein the internal five-dimensional space. Some comments and final remarks are addressed inSection 7. In Appendix A we give our conventions for supersymmetry. N = (0 , holographic backgrounds In this section we shall briefly review the AdS solutions in massive IIA supergravity pre-serving small N = (0 , supersymmetry obtained in [22]. They will constitute our startingpoint from which we will obtain the marginally deformed solutions via a transformationinvolving dualities.The solutions in [22] are of the warped form AdS × S × M , supporting an SU(2)structure on M , equivalently, for these solutions, an SU(3) structure in seven dimensions.Moreover, the five-dimensional space M locally splits into a four-dimensional piece M and an interval. There are two classes of solutions. In this work we will concentrate ona subclass of class I solutions for which M is (conformally) CY . From now on we willconsider CY = T . The NS sector of the solution in the string frame reads ds = u √ h h (cid:18) ds AdS + h h f ds S (cid:19) + (cid:114) h h ds T + √ h h u dρ ,e Φ = 2 h / h / (cid:114) uf , B = f d vol S , (2.1)where the functions u, h , h are functions of ρ only. This is supported with the followingRR field strengths F = h (cid:48) , F = − (cid:18) h − h (cid:48) uu (cid:48) f (cid:19) d vol S , F = (cid:63) F , F = − (cid:63) F ,F = − (cid:18) (cid:18) uu (cid:48) h (cid:19) (cid:48) + 2 h (cid:19) dρ ∧ d vol AdS − h (cid:48) d vol T , F = − (cid:63) F , (2.2)where (cid:48) = ∂ ρ and f = 4 h h + ( u (cid:48) ) , f = 12 (cid:18) − ρ + uu (cid:48) f (cid:19) . (2.3)– 3 –he above background is a supersymmetric solution of massive type IIA supergravity pro-vided h (cid:48)(cid:48) ( ρ ) = 0 , h (cid:48)(cid:48) ( ρ ) = 0 , u (cid:48)(cid:48) ( ρ ) = 0 , (2.4)the first two away from localised sources. The ρ coordinate parametrising the interval canbe taken to be of finite range. This imposes additional constraints on the various functionsof the solution. We require for ≤ ρ ≤ π ( P + 1) that h | ρ =0 = h | ρ =2 π ( P +1) = h | ρ =0 = h | ρ =2 π ( P +1) = 0 . (2.5)The metric functions obeying the above conditions are then explicitly given in Table 1. ≤ ρ ≤ π πj ≤ ρ ≤ π ( j + 1) 2 πP ≤ ρ ≤ π ( P + 1) h ν π ρ µ j + ν j π ( ρ − πj ) µ P − µ P π ( ρ − P π ) h β π ρ α j + β j π ( ρ − πj ) α P − α P π ( ρ − P π ) u b π ρ Table 1 . Piece-wise continuous functions satisfying the conditions in eq. 2.5. The value of u ( ρ ) isthe same in all intervals, as required by supersymmetry. The set of constants ( α j , β j , µ j , ν j , b ) for j = 0 , . . . P parametrising the piece-wisecontinuous functions above are subject to certain constraints imposing continuity of the NSsector along the ρ intervals. The conditions are α k = k − (cid:88) j =0 β j , µ k = k − (cid:88) j =0 ν j . (2.6)The supergravity solution is trustable whenever these constants as well as the number Phave large values. In this section we will construct a family of solutions corresponding to deformations of thesupergravity solutions in eqs (2.1)-(2.2). Such deformations are built upon a sequence ofT dualities and a change of coordinates [43]. The resulting backgrounds are considered tobe holographic duals of the marginally deformed SCFTs dual to the original (undeformed)backgrounds.In order to proceed, we first pick a two-torus in the geometry. For the solution in eq.(2.1) there are two options which will produce inequivalent solutions. They correspond toU(1) ϕ × U(1) x i and U(1) x i × U(1) x j invariant sub-sectors, where ϕ is the azimuthal angleinside the S and x i the coordinates on T . The deformation is achieved by performing aT-duality in one of the coordinates, a shift with parameter λ in the second and T dualityback in the first. The solutions obtained will describe a family of solutions in terms of thefunctions u, h , h and the parameter λ . For other choices where some of these conditions are relaxed see [24, 35]. – 4 –n the first case T : ( ϕ, x ) , following the T duality rules in [45] , the above proceduregenerates the following background ds = u √ h h (cid:18) ds AdS + h h f dθ (cid:19) + (cid:114) h h ( dx + dx + dx ) + √ h h u dρ + 1 f + λ sin θh u (cid:32) u (cid:112) h h sin θdϕ + (cid:114) h h f ( dx − λf sin θdθ ) (cid:33) ,e Φ = 2 h / h / (cid:114) uf + λ h u sin θ , (3.1) B = λh u sin θf + λ h u sin θ ( dx − λf sin θdθ ) ∧ dϕ + f d vol S ,F = h (cid:48) , F = γh u sin θh (cid:48) f + λ h u sin θ ( dx − λf sin θdθ ) ∧ dϕ − (cid:18) h − h (cid:48) uu (cid:48) f (cid:19) d vol S F = − (cid:18) (cid:18) uu (cid:48) h (cid:19) (cid:48) + 2 h (cid:19) dρ ∧ d vol AdS − h (cid:48) d vol T + 12 γ ( h − ρh (cid:48) ) sin θdθ ∧ dx ∧ dx ∧ dx , where the higher fluxes are obtained via the lower ones as indicated in eq (2.2). Thebackground in eq. (3.1) is a solution of massive IIA supergravity if conditions in eq. (2.4)are imposed. We notice the original solution is recovered after turning off the deformationparameter, as expected.For the second case T : ( x , x ) , the procedure outlined above produces the followingbackground ds = u √ h h (cid:18) ds AdS + h h f ds S (cid:19) + √ h h u dρ + (cid:114) h h dx , + (cid:114) h h
11 + λ h h dx , ,e Φ = 2 h / h / (cid:115) u (1 + λ h h ) f , B = f d vol S − λ h h dx ∧ dx λ h h ,F = h (cid:48) , F = − (cid:18) h − h (cid:48) uu (cid:48) f (cid:19) d vol S − λh (cid:48) dx ∧ dx − λ h h (cid:48) h (1 − λ h h ) dx ∧ dx ,F = − (cid:18) (cid:18) uu (cid:48) h (cid:19) (cid:48) + 2 h (cid:19) dρ ∧ d vol AdS − h (cid:48) λ h h d vol T + λ (cid:32) h h (1 + λ h h ) (cid:18) h − uu (cid:48) h (cid:48) f (cid:19) dx ∧ dx + 12 (cid:18) h − uu (cid:48) h (cid:48) f (cid:19) dx ∧ dx (cid:33) ∧ d vol S , (3.2)which is a solution of massive IIA supergravity if conditions in eq. (2.4) are imposed. Wenotice since the S is a spectator subspace for this deformation, we expect N = (0 , supersymmetry will be fully preserved as we will explicitly show in Section 6.– 5 – .1 Quantised charges and brane set-ups In this section we will study the Page charges of the deformed backgrounds. Throughout,we shall use the following definitions Q D p = π ) − p (cid:82) Σ8 − p ˆ f − p , where ˆ f = e − B ∧ f , here f denotes the magnetic (internal) part of the RR polyform F .We start with the solution in eq. (3.1) and consider the following non-trivial cycles ofthe geometry Σ = S , Σ = T , Σ = ( ρ, S ) , Σ = ( S , T ) , Σ (cid:48) = ( x , x , x , θ ) , Σ (cid:48) = ( x , ρ, ϕ ) . (3.3)The Page charges read Q NS = 1(2 π ) (cid:90) Σ H , Q NS (cid:48) = 1(2 π ) (cid:90) Σ (cid:48) H , Q D = 2 πh (cid:48) , Q D = 12 π (cid:90) Σ ˆ f Q D = 1(2 π ) (cid:90) Σ ˆ f , Q D (cid:48) = 1(2 π ) (cid:90) Σ (cid:48) ˆ f , Q D = 1(2 π ) (cid:90) Σ ˆ f . (3.4)If we allow large gauge transformations B → B + πk d vol S , the Page fluxes are those ineq. (3.1) except for ˆ f = −
12 ( h − h (cid:48) ( ρ − πk )) d vol S , ˆ f = 12 ( h − h (cid:48) ( ρ − πk )) d vol S ∧ d vol T . (3.5)The charges in eq. (3.4) computed in πk ≤ ρ ≤ π ( k + 1) are explicitly, Q D = ν k , Q D = µ k , Q D = β k , Q D = α k ,Q D (cid:48) = λ ( kβ k − α k ) Q NS = 1 , Q NS (cid:48) = 0 . (3.6)We notice that for finite ρ ∈ [0 , π ( P + 1)] we have P + 1 parallel NS5 branes. From theabove expressions we see in particular that no extra NS5’ branes were generated by thedeformation. In addition, the above charges are well-defined as long as the set of constants α k , β k , µ k , ν k as well as the combination λ ( kβ k − α k ) , ∈ Z .As we pointed out before, the first two conditions in eq. (2.4) must be satisfied by thesolutions everywhere except at points were we have localised sources. At those points, wehave a change in gradient of the piece-wise linear functions proportional to h (cid:48)(cid:48) , pointingthe possible existence of a source for D p branes via the modified Bianchi identities d ˆ f = j s .From Table 1 we obtain h (cid:48)(cid:48) = P (cid:88) k =1 (cid:18) β k − − β k π (cid:19) δ ( ρ − πk ) , h (cid:48)(cid:48) = P (cid:88) k =1 (cid:18) ν k − − ν k π (cid:19) δ ( ρ − πk ) . (3.7) we will be using units such that g s = α (cid:48) = 1 – 6 –sing this information as well as the Page fluxes of the solution we compute d ˆ f =2 πh (cid:48)(cid:48) dρ, (3.8) d ˆ f = 12 ( ρ − πk ) h (cid:48)(cid:48) dρ ∧ d vol S = 0 , (3.9) d ˆ f = h (cid:48)(cid:48) dρ ∧ (cid:16) d vol T + λ ρ θdθ ∧ dx ∧ dx ∧ dx (cid:17) (3.10) d ˆ f = h (cid:48)(cid:48) ρ − πk ) dρ ∧ d vol S ∧ d vol T = 0 (3.11)where we have used xδ ( x ) = 0 . From this we conclude that D , D (cid:48) as well as D , havingnon-zero sources, correspond to flavour branes whilst D and D are colour ones. Thus inaddition to the D-branes of the seed solution, the deformation has induced (semi-localised)flavour Q D (cid:48) branes. The brane configuration, before the near horizon limit is taken, weargue is associated to the solution above is shown in Table 2. t x r θ φ x x x x ρ N D p D • • • • • • • • • · ν k − − ν k D • • · · · • • • • • µ k D (cid:48) • • • · • • · · · · λk N D D • • • • • · · · · · β k − − β k D • • · · · · · · · • α k NS5 • • · · · • • • • ·
Table 2 . Brane configuration which in the near horizon limit gives the solution in eq. (3.1). Weshow the world-volume directions the branes are suspended as well as their number in the k-thinterval.
For the second solution, we consider the following cycles Σ (cid:48) = ( x , x ) , Σ = T , Σ (cid:48) = ( S , x , x ) , Σ = ( S , T ) , Σ (cid:48) = ( ρ, x , x ) , (3.12)and non-trivial Page forms ˆ f = −
12 ( h − h (cid:48) ( ρ − πk )) d vol S + λ h (cid:48) dx ∧ dx , ˆ f = h (cid:48) d vol T − λ h − h (cid:48) ( ρ − πk )) d vol S ∧ dx ∧ dx , ˆ f = −
12 ( h − h (cid:48) ( ρ − πk )) d vol S ∧ d vol T . (3.13)An analysis as detailed above shows that in addition to the D-branes of the seed solution,the generated Page charges after the transformation ( λ -dependent) are given by Q D (cid:48) = λβ k , Q D (cid:48) = λα k . (3.14)– 7 –his implies the quantisation conditions λβ k ∈ Z , λα k ∈ Z , which requieres rational λ . Inorder to determine if the above charges correspond to colour or flavour branes, we compute d ˆ f = λh (cid:48)(cid:48) dρ ∧ dx ∧ dx + 12 ( ρ − kπ ) h (cid:48)(cid:48) dρ ∧ d vol S (3.15) d ˆ f = h (cid:48)(cid:48) dρ ∧ d vol T − λ ρ − kπ ) h (cid:48)(cid:48) dρ ∧ d vol S ∧ dx ∧ dx . (3.16)Using then (3.7) we find that the effect of the deformation was to add Q D (cid:48) colour and Q D (cid:48) flavour branes respectively. Therefore the original D -NS5-D and D -NS5-D branearrangements are modified by the addition of D (cid:48) branes extended along ( t, x, x , x , ρ ) aswell as semi-localised D (cid:48) branes in ( AdS , S , x , x ) wrapped on T : ( x , x ) . The braneset-up corresponding to this configuration is summarised in Table 3. t x r θ φ x x x x ρ N D p D • • • • • • • • • · ν k − − ν k D • • · · · • • • • • µ k D (cid:48) • • • • • · · • • · λ N D D (cid:48) • • · · · · · • • • λ N D D • • • • • · · · · · β k − − β k D • • · · · · · · · • α k NS5 • • · · · • • • • ·
Table 3 . Brane configuration which in the near horizon limit gives the solution in eq. (3.2). Wealso show the world-volume directions the branes are suspended as well as their number in the k-thinterval. We see the D (cid:48) and D (cid:48) branes are wrapped on T : ( x , x ) . In this section we will study a generalisation to eleven dimensional supergravity of the TsTtransformation studied in the previous section. The seed solutions will be the uplift ofthe background in eq. (2.1)-(2.2) for vanishing Romans mass. The backgrounds obtainedwill correspond to a family of supersymmetric solutions which are out of a subclass of theclassification for AdS eleven dimensional solutions studied in [31].In order to proceed, we consider a vanishing Romans mass in the solution of eq. (2.1)-(2.2) which lead us to consider h = k . The uplift of this solution to eleven dimensions wasfirst constructed in [31]. For latter use we will present some details here. We determine thethree and one-form potentials to be A = (cid:18) uu (cid:48) h + 2 kρ (cid:19) d vol AdS + h (cid:48) x d vol T , A = k θdϕ. (4.1)We notice the 3-form potential above is not globally well-defined. This would be the case if h (cid:48) were a continuous function. Using the usual KK anzats eq. (A.2), the eleven-dimensional– 8 –olution raeds ds =Υ (cid:32) u √ h k ds AdS + (cid:114) h k ds T + √ h ku dρ + k Υ ds S / Z k (cid:33) ,G = dC = d (cid:18)(cid:18) uu (cid:48) h + 2 ρk (cid:19) d vol AdS + 2 k (cid:18) − ρ + 2 πk + uu (cid:48) f (cid:19) d vol S / Z k (cid:19) + h (cid:48) d vol T , (4.2)where ds S / Z k = 14 (cid:32) ds S + (cid:18) k dx + cos θdϕ (cid:19) (cid:33) , Υ = f / √ k (4 u √ h ) / . (4.3)This solution preserves small N = (0 , supersymmetry and supports an SU(2) structure.We will now generalise this class of solutions by performing an SL(3,R) transformation ofcoordinates.For a solution which is SL(3,R) invariant we use the anzats ds = ∆ − / g µν dx µ dx ν + ∆ / M ab D φ a D φ b ,C = C (0) D φ ∧ D φ ∧ D φ + 12 C (1) ab ∧ D φ a ∧ D φ b + C (2) a ∧ D φ a + C (3) , D φ a = dφ a + A aµ dx µ , (4.4)where the a, b indices correspond to the three-torus directions, g µν is the transverse eightdimensional metric and detM=1. We have two possible choices for which we can apply thetransformation. Namely T : ( x , x , x ) and T : ( x , x , x ) .In the first case, the background in eq. (4.2) can be bring into the form of eq. (4.4)provided we identify A aµ = 0 , M ab = ∆ − / (cid:18) h f u (cid:19) / δ ab , ∆ = h f uC (0) = − h (cid:48) x , C (3) = − (cid:18) uu (cid:48) h + 2 kρ (cid:19) d vol AdS + B ∧ dx ,C (1) ab = C (2) a = 0 , (4.5) ∆ − / g µν dx µ dx ν = √ k (cid:18) f √ h u (cid:19) / (cid:32) u √ h k (cid:18) ds AdS + h kf ds S (cid:19) + (cid:114) h k dx + √ h ku dρ (cid:33) + k (cid:18) u h f (cid:19) / ( 2 k dx + cos θdϕ ) , We then use the transformation rules spelled out in [44] to obtain the new backgroundparametrised by λ . The transformation for the one-form A a , using (4.5), gives ˜ A a = A a + λ(cid:15) abc C (1) bc = 0 and therefore ˜ D φ a = D φ a = dφ a . On the other hand, the non-trivialtransformation associated to τ = − C (0) + i ∆ / reads ˜ τ = τ / (1 + λτ ) , from which we obtain ˜∆ = G ∆ , ˜ C (0) = G (cid:16) C (0) − λ ( C + ∆) (cid:17) , G = (1 + 2 λC (0) − λ ( C + ∆)) − . (4.6)– 9 –he deformed background then reads ds = G − / (cid:34) Υ (cid:32) u √ h k ds AdS + (cid:114) h k dx + √ h ku dρ + k Υ ds S / Z k (cid:33) + G Υ (cid:114) h k ds T (cid:35) ,G = dC , (4.7)where C = − (cid:18) uu (cid:48) h + 2 kρ + λ x h (cid:0) h f + uu (cid:48) h (cid:48) (cid:1)(cid:19) d vol AdS + ˜ C d vol T + 4 k (cid:18) f − λ x (cid:18) h − uu (cid:48) h (cid:48) f (cid:19)(cid:19) d vol S / Z k . (4.8)This background is a solution of 11d supergravity when conditions in eq. (2.4) are imposed,and reduces to the undeformed solution for λ = 0 , as expected.In the second case T : ( x , x , x ) , the solution obtained following the procedurespelled out above corresponds to the uplift to eleven dimensions of the solution in eq. (3.2).The eleven-dimensional background reads ds =Υ(1 + λ h h ) / (cid:32) u √ h c ds AdS + h h f ds S + √ h h u dρ + (cid:114) h h ( dx + dx )++ (cid:114) h h
11 + λ h h ( dx + dx ) + c (cid:32) ds S + Dy λ h h (cid:33)(cid:33) ,G = − (cid:18) ∂ ρ (cid:18) uu (cid:48) h (cid:19) + 2 c (cid:19) dρ ∧ d vol AdS − h (cid:48) λ h c d vol T − λ h (cid:48) (cid:16) λ h c (cid:17) dρ ∧ dx ∧ dx ∧ Dy + λ (cid:32) h λ h c dx ∧ dx + (cid:18) h − uu (cid:48) h (cid:48) f (cid:19) dx ∧ dx (cid:33) ∧ d vol S + c ∂ ρ f dρ ∧ d vol S ∧ Dy, (4.9)where Dy = 2 c dy + cos θdϕ − c λx h (cid:48) dx , (4.10)and Υ was defined in eq. (4.3). This background is a solution of 11d supergravity whenconditions in eq. (2.4) are imposed. In Section 6 we will show that the solutions presentedin this section preserve N = (0 , supersymmetry supporting an identity structure.Before to close this section, it is worth noticing that the solutions in eqs. (4.7) and(4.9) can be used as seed solutions in order to generate other families of supersymmetricsolutions. For instance, after appropriate analytical continuations we can generate solutionswith AdS / Z k × S factors which further reduction to IIA along the Hopf-fibre directionof AdS will generate new AdS × S solutions in IIA supergravity, which can be furtherextended to massive IIA, generalising those studied in [31], etc.– 10 – Holographic central charge
The main goal of this section will be to compute the central charge characterising the newfamily of solutions. For the seed solutions this was done in [24, 31] and using the analysisof the spin-2 spectrum in [29]. A generic result involving the deformations discussed aboveis that they leave the internal space volume transverse to AdS -weighted by the dilaton-invariant. We then anticipate the central charges will be the same before and after thedeformation.In order to see this explicitly, we consider the metric of the solutions written in thefollowing way ds = a ( r, (cid:126)y ) (cid:0) dx ,d + b ( r ) dr (cid:1) + g ij ( r, (cid:126)y ) dy i dy j , (5.1)where x ,d parametrises M ,d Minkowski space and g ij the metric of the internal space.The holographic central charge is then given by the following expression [46] c hol = d d G N b ( r ) d/ H d +12 H (cid:48) d , (5.2)where H = (cid:18)(cid:90) d(cid:126)y (cid:113) e − det ( g ij ) a d (cid:19) . (5.3)For the ten dimensional solution, since the deformations acted on the internal space of thesolutions, we clearly see the quantities a ( r, (cid:126)y ) , b ( r ) in eq. (5.1) are spectator under thedeformations. In addition, we find that e ˜Φ = e Φ (cid:113) λ det ( g T ) , ˜ g T = g T λ det ( g T ) (5.4)where tilde denotes fields after the deformation. It is then easy to see that e − det (˜ g ij )˜ a d = e − det ( g ij ) a d , giving ˜ c hol = c hol as anticipated. This result goes through for the eleven-dimensional solutions after considering the relation between the ten and eleven-dimensionalquantities in the KK anzats (A.2) and H = (cid:16)(cid:82) d(cid:126) ˆ y (cid:112) det (ˆ g ij )ˆ a d (cid:17) , where quantities with hatare eleven-dimensional ones.After we have characterised the backgrounds by computing their central charges, thegoal is to compare them with the central charges obtained from the putative dual fieldtheories to these solutions, in the holographic limit. Some comments are in order. Forinstance, in the case of the field theory read off from the brane configuration in Table 3,we can achieve an anomaly free quiver field theory following the rules in [23, 24]. Never-theless, this gives a central charge that is apparently changing due to the extra gauge andflavour group insertions. We would expect cancelations among them that will give the samecentral charge as before the deformation, or that their contributions are sub-leading in theholographic limit. We will elaborate more on this in a forthcoming publication.– 11 – Comments on supersymmetry and G-structure of the solutions
In this section we will study the supersymmetries preserved by the supergravity solutionsin eqs. (3.1), (3.2) and (4.7), (4.9), based on the explicit form of the Killing spinors ofthe original solution (2.1). The conventions we follow for supersymmetry are detailed inAppendix A. The solution in eq. (2.1) preserves small N = (0 , supersymmetry byconstruction. In the conventional approach, this implies the existence of two algebraicconditions on the ten-dimensional Majorana-Weyl (MW) spinor ensuring the vanishing ofthe supersymmetry variations.In order to see this explicitly, we decompose the ten-dimensional gamma matrices asfollows Γ α = σ ⊗ ρ α ⊗ I , Γ µ = σ ⊗ I ⊗ γ µ , (6.1)where ρ α and γ µ are three and seven-dimensional gamma matrices respectively and the σ i are the usual Pauli matrices. In this notation the chirality matrix is Γ = − σ ⊗ I ⊗ I .After plugging the solution in eq. (2.1) (in the natural frame) into eq. (A.11) we find theMW Killing spinor takes the form (cid:15) = (cid:32) (cid:33) ⊗ ζ ⊗ χ , (cid:15) = (cid:32) (cid:33) ⊗ ζ ⊗ χ , (6.2)where ζ is the AdS Killing spinor and χ , = e A e i θ σ γ e ϕ γ e − arctan 2 √ h h u (cid:48) γ σ χ (0)1 , , A = 12 log u √ h h , (6.3)a seven-dimensional spinor satisfying the projection conditions γ χ , = − χ , , √ f (cid:16) (cid:112) h h γ σ + u (cid:48) γ iσ (cid:17) χ , = χ , (6.4)where , , and . . . are flat indices corresponding to the ρ , S and T directionsrespectively. The purpose of this section, is to find the number of spinor componentswhich are compatible with the TsT transformation.Since the deformation involves a sequence of T dualities, a condition for preservingKilling spinors reduces to their invariance by the action of the Kosmann-Lie derivativealong the Killing vector K associated to the isometric direction we picked to perform theduality L K (cid:15) = 0 , where (cid:15) is the Killing spinor of the un-dualised solution. By considering K = ∂ y , the above condition reduces to ∂ y (cid:15) = 0 [47]. Moreover, invariance under thechange of coordinates in the second direction also requires independence of it on the spinor.Therefore, supersymmetry is compatible with TsT transformations as long as the spinor isuncharged under the directions used for the transformation [48].For the first solution in eq. (3.1), there is a residual U(1) ϕ which we may think ofas a candidate R-charge for N = (0 , preserved supersymmetry. However, the spinors(6.2) are charged under this coordinate and T duality along this direction will project outthis dependence. The residual U(1) ϕ is therefore a global symmetry and supersymmetryis completely broken. In other words, compatibility with the TsT transformation imposes– 12 –he projection condition γ χ , = 0 breaking all supersymmetries. Despite the breaking ofsupersymmetries, this solution is interesting in its own since it still solves the BPS conditionin eq. (2.4).For the second marginally deformed solution in eq. (3.2), the spinor is independentof the T directions, so we ensure supersymmetry is fully preserved. To be more precise,working with the supersymmetry transformations for the solution in eq. (3.2), we find δ ˜ λ = e − arctan λ (cid:113) h h γ δλ , δ ˜ λ = δλ ,δ ˜ ψ µ = e − arctan λ (cid:113) h h γ δψ µ , δ ˜ ψ µ = δψ µ , (6.5)where tilde denotes fields after the transformation, provided we identify ˜ χ = e − arctan λ (cid:113) h h γ χ , ˜ χ = χ , (6.6)ensuring supersymmetry is preserved as the original solution does. This is along the linesof the generic result in [48], which in addition showed that the entire information of thetransformation is encoded in an antisymmetric bi-vector associated to classical r-matricessolving the Yang-Baxter equation.Let us now turn to the G-structure characterising the above background. To begin with,the solutions in [24] were constructed by imposing that they support an SU(2) structureon the five-dimensional internal space M transverse to AdS × S . For these solutions,this implies that the internal five-dimensional spinors are globally parallel. The deformedKilling spinors break the above condition, each of which defining an SU(2) structure, theintersection of which gives an identity structure. To be more precise, given the rotationof the internal spinor under TsT eq. (6.6), the transformed MW spinor takes the form(6.2) with the internal spinor transformed accordingly χ , → ˜ χ , . In addition, the seven-dimensional spinor can be further decomposed into S × M factors according to eq. (6.3).Namely, χ , = e A ξ ⊗ η , , where ξ is a Killing spinor on the S charged under SU(2) R . AnSU(2) structure on M implies η = η, η = η. (6.7)Using (6.6) the TsT MW spinors are given by (cid:15) = (cid:32) (cid:33) ⊗ ζ ⊗ ˜ χ , (cid:15) = (cid:32) (cid:33) ⊗ ζ ⊗ ˜ χ , (6.8)where, using the 2+5 split of the internal spinor χ , we find ˜ η = 1 (cid:113) λ h h ( η − λ (cid:114) h h ( γ (5) η )) , ˜ η = η, (6.9) In a common basis the spinors can be written as η = η and η = aη + bη c + c ωη , where ω is a complexone-form. The case of SU(2) structure sets c = 0 . In addition, without loss of generality, we can choose forthese solutions b = 0 . class I solutions are further characterised by a =1. – 13 –herefore the spinors ˜ η , are nowhere parallel defining a point-dependent SU(2) × SU(2)structure, that we will refer to it as dynamical. This is then described in terms of thelargest common subgroup, which then defines a dynamical identity structure. Notice wecould have also analysed the G-structure of the solution in terms of the seven-dimensionalspinor. In this case the seed solution supports an SU(3) structure. It would then be possibleto understand the fate of the seven-dimensional G-structure following [49]. The analysisof this section suggest this may give a dynamical SU(3) structure, and will provide a newexample of AdS solutions with dynamical SU(3) structure. We plan to report on this in aforthcoming publication.For the eleven dimensional solutions in eqs. (4.7) and (4.9) the preserved MajoranaKilling spinors can be ascertained just as we did for the ten dimensional case. Namely, thepreserved Killing spinors are those which are independent of the directions along which weperformed the transformation. Using the relation between the eleven and ten-dimensionalspinors (A.8) together with (A.4) we easily see that small N = (0 , supersymmetry ispreserved. Once again whenever the deformation parameter is turned offƒ we recover theundeformed Majorana Killing spinor defining an SU(2) structure. In the case at hand wehave a dynamical identity structure instead. In this paper we have presented new solutions in massive IIA and eleven-dimensional su-pergravity obtained via TsT transformations and the analog in eleven dimensions. Thesolutions obtained preserve small N = (0 , supersymmetry and support a dynamicalidentity structure on the five-dimensional internal submanifold of the solution, as long aswe do not use the azimuthal angle inside the S in the procedure. The new backgrounds inten and eleven dimensional supergravity constitute a whole family of solutions parametrisedby the deformation parameter λ and linear functions satisfying the conditions in eq. (2.4).To the best of our knowledge, a complete classification of these solutions is still missing inthe literature. One can in principle follow the same procedure as the one outlined in [22]for the SU(2) structure case. That is to say construct bispinors out of seven-dimensionalspinors supporting a (dynamical) identity structure in the internal five-dimensional spaceand obtain geometrical constraints in the form of the solution from the differential condi-tions implied by supersymmetry. Moreover, in terms of seven-dimensional G-structure, theseed solutions support an SU(3) structure. The analysis we followed in Section 6 suggeststhis becomes a dynamical SU(3) structure after the transformation. Progress on classifica-tion of AdS geometries supporting a dynamical SU(3) structure was recently reported in[37].For the ten-dimensional solutions, we studied the Page charges and associated braneconfigurations. We showed that depending on the two-torus chosen, the deformation addseither colour or flavour branes or both to the seed configuration. Holographically, Thebackgrounds obtained correspond to marginal deformations of the SCFT dual to the seedsolutions. We verified this by computing the central charge of the deformed backgrounds,showing they are the same before and after the transformation. In the field theory side– 14 –ide, we can engineer a dual quiver quantum field theory with the information obtainedfrom the Hannany-Witten brane set-ups associated to the solutions. The specification ofthe dual quantum field theories and more field theory aspects of the solutions are left for aforthcoming publication. Acknowledgments
I am indebted to Carlos Núñez for many useful discussions. I also thank Yolanda Lozano,Niall Macpherson, Anayeli Ramirez and Stefano Speziali for comments and correspondence.
A Supersymmetry conventions
In this appendix we will set the conventions for supersymmetry. We find useful to reviewhow to obtain the ten-dimensional supersymmetry variations from the eleven dimensionalone by dimensional reduction.Let us start with the supersymmetry variation for the gravitino in eleven-dimensionalsupergravity. It is δ ˆ ψ M = (cid:20) ∇ M + 1288 (cid:16) ˆΓ N ...N M − δ N M ˆΓ N N N (cid:17) ˆ F N ...N (cid:21) ˆ (cid:15), (A.1)where ∇ M = ∂ M + ω BCM ˆΓ BC . We will study the reduction of the above supersymmetryvariation along x = z . From now on objects with hat will denote eleven dimensionalquantities. To proceed, we use the usual ansatz for the string frame metric and three-formpotential, which read ds = = η AB E AN E BN dx N dx N = e − Φ ds + e Φ ( dz + A ) ,C = A + B ∧ dz, (A.2)where Φ is the dilaton and M, N = ( µ, z ) , A = ( a, z ) are curved and tangent space indicesrespectively. The spin-connection components of the above geometry are thus given by ˜ ω ab = e Φ3 ( ω ab − η c [ a ∂ b ] φe c ) − e Φ F ab e z , ˜ ω za = 23 e Φ3 ∂ a Φ e z + 12 e Φ F ab e b , (A.3)where F = 2 ∂A . The dimensional reduction of the eleven-dimensional gravitino ˆ ψ M generates a ten-dimensional gravitino and the dilatino as follows ˆ ψ µ = e Φ6 ( ψ µ −
16 ˆΓ µ λ ) , ˆ ψ z = 13 e Φ6 ˆΓ z λ. (A.4)In the same vein, the ˆ F = 4 ∂C field strength contains two pieces the components ofwhich are ˆ F αβγδ and ˆ F αβγz . Using flat indices we identify ˆ F abcd = E N a E N b E N c E N d ˆ F N N N N = 4 e φ (cid:0) ∂ [ a A bcd ] − A [ a H bcd ] (cid:1) = e Φ F abcd , ˆ F abcz = E N a E N b E N c E zz ˆ F N N N z = e Φ3 H abc , (A.5)– 15 –here H = 3 ∂B . Moreover, the dimensional reduction of eq. (A.1) generates the terms δ ˆ ψ z = ∂ z ˆ (cid:15) + 14 (cid:16) ω z ab ˆΓ ab + 2 ω z za ˆΓ za (cid:17) ˆ (cid:15) + 1288 (cid:16) ˆΓ ABCDz − δ Az ˆΓ BCD (cid:17) ˆ F ABCD = 13 e Φ / ˆΓ z (cid:18) /∂φ − · e Φ /F ˆΓ z − /H ˆΓ z + 14 · e Φ /F (cid:19) (cid:15), (A.6) δ ˆ ψ a = E µa ∂ µ ˆ (cid:15) + 14 ω aBC ˆΓ BC ˆ (cid:15) + 1288 (cid:16) ˆΓ BCDEa − δ Ba ˆΓ CDE (cid:17) ˆ F BCDE ˆ (cid:15) = e Φ6 (cid:18) ( e µa ∂ µ + 14 ω abc ˆΓ bc − ∂ a Φ −
16 ˆΓ ca ∂ c Φ + 14 e Φ F ac ˆΓ cz ) (A.7) + 1288 e Φ ˆΓ a /F − e Φ ˆΓ bcd F abcd + 172 ˆΓ a /H ˆΓ z − /H µ ˆΓ z (cid:19) (cid:15), where we have introduced the notation /F = F µ ...µ n Γ µ ...µ n and the relation between elevenand ten dimensional spinors ˆ (cid:15) = e − Φ6 (cid:15). (A.8)We then decompose the eleven-dimensional Gamma matrices in terms of the ten-dimensional ones in the following way ˆΓ a = Γ a σ , ˆΓ z = σ , a = 1 , . . . , (A.9)which is related to the decomposition of the ten-dimensional Majorana spinor into itschiral components (cid:15) = (cid:32) (cid:15) (cid:15) (cid:33) , (A.10)satisfying Γ (cid:15) = − σ (cid:15) . Using (A.4)-(A.9) we then identify the supersymmetry variationsfor the dilatino and gravitino. For non-zero Romans mass, the expressions obtained can beslightly generalised to δλ = /∂ Φ (cid:15) − · /Hσ (cid:15) + e Φ (cid:18) F σ + 32! /F iσ + 14! /F σ (cid:19) (cid:15),δ Ψ µ = ∇ µ (cid:15) − /H µ σ (cid:15) + e Φ (cid:18) F σ + 12! /F iσ + 14! /F σ (cid:19) Γ µ (cid:15). (A.11) References [1] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Int.J. Theor. Phys. , 1113-1133 (1999), arXiv:9711200[2] A. Strominger and C. Vafa, “Microscopic origin of the Bekenstein-Hawking entropy,” Phys.Lett. B , 99-104 (1996), arXiv:9601029[3] J. M. Maldacena, A. Strominger and E. Witten, “Black hole entropy in M theory,” JHEP ,002 (1997), arXiv:hep-th/9711053[4] R. Minasian, G. W. Moore and D. Tsimpis, “Calabi-Yau black holes and (0,4) sigma models”Commun. Math. Phys. , 325-352 (2000), arXiv:hep-th/9904217 – 16 –
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