aa r X i v : . [ h e p - t h ] N ov Half BPS states in
AdS × Y p,q Edi Gava a,b,c , Giuseppe Milanesi b,c,d ,K.S. Narain a and Martin O’Loughlin e a High Energy Section, The Abdus Salam International Centre for Theoretical Physics,Strada Costiera 11, 34014 Trieste, Italy b Istituto Nazionale di Fisica Nucleare, Sezione di Trieste c Scuola Internazionale Superiore di Studi Avanzati,Via Beirut 2-4, 34014 Trieste, Italy d Institut f¨ur Theoretische Physik,ETH Z¨urich, CH-8093 Z¨urich, Switzerland e University of Nova Gorica,Vipavska 13, 5000, Nova Gorica, Slovenia
Abstract
We study a class of solutions of IIB supergravity which are asymptotically
AdS × Y p,q . They have an R × SO (4) × SU (2) × U (1) isometry and preserve half of the 8 su-percharges of the background geometry. They are described by a set of second orderdifferential equations that we have found and analysed in a previous paper, wherewe studied 1/8 BPS states in the maximally supersymmetric AdS × S background.These geometries correspond to certain chiral primary operators of the N = 1 super-conformal quiver theories, dual to IIB theory on AdS × Y p,q .We also show how to recover the AdS × Y p,q backgrounds by suitably doublingthe number of preserved supersymmetries. We then solve the differential equationsperturbatively in a large AdS radius expansion, imposing asymptotic AdS × Y p,q boundary conditions. We compute the global baryonic and mesonic charges, includingthe R-charge. As for the computation of the mass, i.e. the conformal dimension ∆of the dual field theory operators, which is notoriously subtle in asymptotically AdS backgrounds, we adopt the general formalism due to Wald and collaborators, whichgives a finite result, and verify the relation ∆ = 3 R/
2, demanded by the N = 1superconformal algebra. SISSA 50/2007/EP
Introduction
One of the most impressive checks of the
AdS/CF T has been obtained a few years ago[1], where a very precise correspondence between supergravity geometries and states in thedual SU ( N ) N = 4 Yang-Mills theory on R × S has been established at the 1/2 BPSlevel. More precisely, the free-fermion picture arising in the large N gauge theory reducedon S and restricted to the 1/2 BPS sector, has been shown to appear quite precisely inthe exact solution of the 1/2 BPS geometries on the supergravity side. This goes beyondthe giant graviton regime, which corresponds to probe D3 branes wrapped on S ’s eitherin AdS or S [2, 3, 4, 5] , in the sense that it captures the full gravitational backreactedgeometry. Attempts to generalise this picture to less supersymmetric geometries/statesappeared recently in [6, 7, 8, 9, 10, 11, 12]. An important class of non-local normalizablestates (Wilson lines) and the corresponding dual geometries were studied in [13, 14]Of course, another, but related, direction to explore would be to consider BPS states inless supersymmetric bulk theories. Interesting examples are the dual pairs given by stringtheory on AdS × Y p,q and certain N = 1 Superconformal Quiver Gauge Theories, whichhave been subject of intense study recently. In [15, 16] the explicit metric on a class ofSasaki Einstein manifolds Y p,q was constructed. A direct generalisation of the AdS/CF T correspondence relates Type IIB String Theory on
AdS × Y p,q , with N = 1 Quiver GaugeTheories [17]. The parameters are identified as follows L AdS πℓ s = (cid:18) λ π π V ol ( Y p,q ) (cid:19) g s = λN . (1.1)Every Y p,q manifold has an SU (2) × U (1) × U (1) isometry group and the AdS × Y p,q solutions preserve 8 of the original 32 supersymmetries of type IIB supergravity. Super-symmetric branes wrapping cycles in Y p,q have been analysed in the probe approximationin [18, 19] and they may be considered as generalisations of giant gravitons. Dual giantgravitons were studied in [21, 22]. A distinguishing feature of the Y p,q manifolds, unlike S , is the presence of a non-trivial 3-cycle. D3-branes can thus wrap such a non trivialcycle and be stable: such branes are dual to baryons in the gauge theory, the so calleddibaryons, which are built out of products of N chiral superfields [20]. Correspondingly,on the supergravity side there is a gauge field coming from the four-form Ramond-Ramondgauge field, which is dual to the baryonic current of the Gauge Theroy.In the quiver gauge theories associated to Y p,q manifolds, there are 2 p SU ( N ) gaugegroups and 4 types of chiral superfields, X , Y , U i and V i , i = 1 , SU ( N ) × SU ( N ), with the precise gauge assignments encoded in the quiver diagram. Thefields U and V are furthermore doublets of an SU (2) flavour symmetry. With a genericsuperfield A βα , α ∈ N and β ∈ ¯ N , in the bifundamental of SU ( N ) × SU ( N ), one canconstruct dibaryonic gauge singlets ǫ α ,...,α N ǫ β ,...,β N A α β · · · A α N β N The dibaryons constructedwith the SU (2) doublets U i and V i are furthermore in the N + 1 dimensional representa-tion of SU (2). In addition to baryonic-like operators one can construct also mesonic-likeoperators, which are neutral under the baryonic charge. These are the precise analogs2f giant gravitons of the N = 4 theory. In any case, since our geometries preserve an SU (2), in addition to R × SO (4) × U (1), they correspond to SU (2) singlet operators onthe gauge theory side, e.g. those constructed with the chiral superfields X and Y . Thethree U (1) charges, i.e. the R-charge, a flavour U (1) and the baryonic charge, will appearas integration constants in our asymptotic solutions.In [12] solutions of the type IIB equations of motion with non trivial R-R 5-form and R × SO (4) × SU (2) × U (1) isometry group preserving 4 supercharges have been studied. AdS × Y p,q geometries are clearly contained in this class: the R × SO (4) is the non compactversion of U (1) × SO (4) ⊂ SO (2 , SU (2) × U (1) × U (1) isometry group of Y p,q is contained in the generic SU (2) × U (1) bosonic symmetry.In this paper we first show in detail how to recover the AdS × Y p,q geometries from thegeneric solutions studied in [12] by requiring that additional 4 supercharges be preserved.We then study 1/2 BPS excitations of such geometries, namely generic 1/8 BPS solutionsof type IIB supergravity with AdS × Y p,q asymptotics and R × SO (4) × SU (2) × U (1)isometry: they represent an expansion of the fully backreacted geometries of D3 branes in AdS × Y p,q . The brane source is substituted by flux in the same spirit as in the original[1]. Such geometries carry three net global U (1) charges which are dual to the R-charge, a U (1) flavour charge and the baryonic charge of the gauge theory. They are determined byfour scalar functions defined on a halfspace which solve four nonlinear coupled differentialequations. In order to specify the asymptotics and charges of the solutions we solve suchequations perturbatively at large AdS radius. The zeroth order fixes the metric and theRR 5-form as needed to describe correctly the AdS × Y p,q geometries, the first subleadingcorrections determine the aforementioned global U (1) charge and the second subleadingcorrection is necessary to obtain the value of the mass. Solutions which carry only R -chargehave been studied in [23] at the linearised level.The definition of mass is somewhat subtle in asymptotically AdS spacetimes, [24, 25]but it is even subtler when one is dealing with states in asymptotically
AdS × X , withcompact X , due to the fact that the subleading terms in the metric, that in principlecan be used to determine the mass, mix the AdS and M coordinates. We deal with thisproblem by adopting a 10-dimensional version of the general construction of [26], to findthe conserved Hamiltonian and thus the correct definition of the mass. We then determinethe mass of our states and check that the BPS condition, relating the mass to the R-charge,is indeed satisfied by our asymptotic solutions.The paper is organised as follows. In Section 2 we give a brief summary of the resultsof [12]. In Section 3 we show how to obtain the AdS × Y p,q geometries from the generalsolutions. In Section 4 we solve the system of differential equations up to second orderin large AdS radius (the details of the second order solutions are showed in AppendixA). In Section 5 we show how to obtain the R charge and the U (1) flavour charge of thesolutions. In Section 6 we discuss subleading corrections to the RR 5-form and derive thebaryon charge of the solutions. In Section 7 we discuss how to correctly define the mass fora space-time which is asymptotically a product with an AdS factor. Finally, in Section 8we present some conclusions. 3 Description of 1/8 BPS States
Generic solutions of type IIB Supergravity preserving 4 of the 32 supersymmetries ofthe theory and an R × SO (4) × SU (2) × U (1) bosonic symmetry have been constructedperturbatively in [12]. The metric takes the formd s = − h − (d t + V i d x i ) + h ρ ρ ( T δ ij d x i d x j + d y ) + ˜ ρ d ˜Ω ++ ρ (cid:0) ( σ ˆ1 ) + ( σ ˆ2 ) (cid:1) + ρ ( σ ˆ3 − A t d t − A i d x i ) (2.1)with i = 1 ,
2; the coordinate y is the product of two of the radii, y = ρ ˜ ρ > . (2.2)and the function h is given by h − = ˜ ρ + ρ (1 + A t ) . (2.3)The space is a fibration of a squashed 3-sphere (on which the SU (2) left-invariant 1-forms σ ˆ a are defined) and a round 3-sphere ˜Ω (on which the SU (2) left-invariant 1-forms σ ˜ a aredefined) over a four dimensional manifold.The left invariant 1-forms are given by: σ ˆ1 = − (cos ˆ ψ d ˆ θ + sin ˆ ψ sin ˆ θ d ˆ φ ) σ ˜1 = − (cos ˜ ψ d ˜ θ + sin ˜ ψ sin ˜ θ d ˜ φ ) σ ˆ2 = − ( − sin ˆ ψ d ˆ θ + cos ˆ ψ sin ˆ θ d ˆ φ ) σ ˜2 = − ( − sin ˜ ψ d ˜ θ + cos ˜ ψ sin ˜ θ d ˜ φ ) σ ˆ3 = − ( d ˆ ψ + cos ˆ θ d ˆ φ ) σ ˜3 = − ( d ˜ ψ + cos ˜ θ d ˜ φ ) (2.4)and satisfy the relations (with σ a being either σ ˆ a or σ ˜ a )d σ a = ǫ abc σ b ∧ σ c . (2.5)With this normalisation the metric on the unit radius round three sphere is given by d Ω = ( σ ) + ( σ ) + ( σ ) . (2.6)The only non trivial field strength in our Ansatz is the Ramond-Ramond 5-form: it ismore conveniently expressed in terms of the “d-bein” e = h − (d t + V i d x i ) (2.7) e j = h ρ ρ T δ ji d x i (2.8) e = h ρ ρ d y (2.9) e ˆ a = ( ρ σ ˆ a ˆ a = 1 , ρ ( σ ˆ3 − A µ d x µ ) ˆ a = 3 (2.10) e ˜ a = ˜ ρσ ˜ a (2.11)4s F (5) = 2 (cid:16) ˜ G mn e m ∧ e n + ˜ V m e m ∧ e ˆ3 + ˜ ge ˆ1 ∧ e ˆ2 (cid:17) ∧ ˜ ρ d ˜Ω +2 (cid:16) − G pq e p ∧ e q ∧ e ˆ1 ∧ e ˆ2 ∧ e ˆ3 + ⋆ ˜ V ∧ e ˆ1 ∧ e ˆ2 − ⋆ ˜ g ∧ e ˆ3 (cid:17) , (2.12)where G mn = 12 ǫ mnpq ˜ G pq (2.13) ⋆ ˜ V = 13! ǫ mnpq ˜ V m e n ∧ e p ∧ e q (2.14) ⋆ ˜ g = ˜ ge ∧ e ∧ e ∧ e . (2.15)The complete solution can be expressed in terms of four independent functions m, n, p, T defined on the halfspace ( x , x , y ), as follows ρ = mp + n m y ρ = p m ( mp + n ) ˜ ρ = mmp + n h = mp mp + n A t = n − pp A i = A t V i − ǫ ij ∂ j ln T (2.16)and d V = − y ⋆ [d n + ( nD + 2 ym ( n − p ) + 2 n/y )d y ] (2.17) ∂ y ln T = D (2.18) D ≡ y ( m + n − /y ) , (2.19)where ⋆ indicates the Hodge dual in the three dimensional diagonal metricd s = T δ ij d x i d x j + d y . (2.20)The various four-dimensional forms from which the 5-form field strength is constructedare given by ˜ g = 14 ˜ ρ (cid:20) − ρ ρ (1 + A t ) (cid:21) (2.21)˜ V = 12 1 ρ ˜ ρ d(˜ gρ ˜ ρ ) (2.22) Gρ ρ = d B t ∧ (d t + V i d x i ) + B t d V + d ˆ B (2.23)˜ G ˜ ρ = 12 gρ ˜ ρ d A + d ˜ B t ∧ (d t + V i d x i ) + ˜ B t d V + d ˆ˜ B , (2.24)with ˜ B t = − y n − /y p d ˆ˜ B = − y ⋆ [d m + 2 mD d y ] B t = − y nm d ˆ B = 116 y ⋆ [d p + 4 yn ( p − n )d y ] . (2.25)5he Bianchi identities on F (5) and the integrability condition for (2.17) give three secondorder differential equations on m, n, p which, together with (2.18) give a system of nonlinearcoupled elliptic differential equations y ( ∂ + ∂ ) n + ∂ y (cid:0) y T ∂ y n (cid:1) + y ∂ y (cid:2) T (cid:0) yDn + 2 y m ( n − p ) (cid:1)(cid:3) + 4 y DT n = 0 y ( ∂ + ∂ ) m + ∂ y (cid:0) y T ∂ y m (cid:1) + ∂ y (cid:0) y T mD (cid:1) = 0 y ( ∂ + ∂ ) p + ∂ y (cid:0) y T ∂ y p (cid:1) + ∂ y (cid:2) y T ny ( n − p ) (cid:3) = 0 ∂ y ln T = D . (2.26) AdS × Y p,q solutions Taking any solution described in Section 2 and assuming rotational symmetry in the { x , x } plane, the bosonic symmetry is enhanced to R × SO (4) × SU (2) × U (1) × U (1).We will first consider a subset of solutions which preserve 8 supersymmetries (the genericsolution preserves only 4 of them as explained in the previous section). The well known AdS × Y p,q [16] are clearly contained in this subset: the round S is a factor in AdS , assuggested by the analysis in [12], with R × SO (4) the non compact version of U (1) × SO (4) ⊂ SO (2 , SU (2) × U (1) × U (1) is the isometry group of the generic Y p,q metric. Since the solutions described in [12] generically preserve only 4 supersymmetries, the
AdS × Y p,q geometries will be specified by a set of constraints on the four functions m, n, p and T . We will now show how these constraints arise.The supersymmetry parameters that leave invariant our background are the solutionsto the Killing spinor equation δχ M = ∇ M ψ + i480 F M M M M M Γ M M M M M Γ M ψ = 0 . (3.1)As a consequence of the symmetry assumptions we look for a solution ψ of the form ψ = ε ⊗ ˆ χ ⊗ ˜ χ ( b ) . (3.2)Here ε is an 8 component complex spinor and ˆ χ, ˜ χ ( b ) are 2 component complex spinorsdefined on the two 3-spheres satisfying ∂∂ω ˆ a ˆ χ = 0 σ ˆ3 ˆ χ = s ˆ χ (3.3) ∇ ′ ˜ a ˜ χ ( b ) = b i2 σ ˜ a ˜ χ ( b ) (3.4)where ω ˆ a , ω ˜ a are coordinates on the two spheres. ∇ ′ is the covariant derivative on the unitradius three sphere and s, b = ±
1. The spinor ˆ χ is a singlet under the SU (2) L isometry6f the squashed 3-sphere, while the spinors ˜ χ ± transform as the (0 , ) for upper sign and( ,
0) for the lower sign, of the SO (4) isometry of the round S , which is part of AdS .The analysis in [12] fixes b = s = 1, i.e. ˜ χ has definite chirality in SO (4) and ˆ χ ishighest weight of the broken SU (2) R . ε is proportional to some ε obeying ε † ε = 1. Sincewe have a doublet of ˜ χ (1) , the space of solutions is 2 dimensional and complex giving rise to4 real preserved supersymmetries. We will show that AdS × Y p,q geometries are obtainedby requiring that spinors with b = − s = 1 are also solutions of the equations (3.1). Inthis case there are two doublets of ˜ χ and thus 8 real solutions to (3.1). This agrees withwhat one expects from the N = 1 SCFT side: there, out of the 4 pairs of Killing spinors ξ A ± , A = 1 , . . . , of SU (4) of the N = 4 theory on R × S , obeying D µ ξ A ± = ± i σ µ ξ A ± (3.5)only the SU (3) singlet ξ ± in SU (3) × U (1) ⊂ SU (4) survives in the N = 1 case. This has SU (4) weights ( , , ) and, picking the SO (4) inside SU (4) corresponding for exampleto the first two entries, we see that it is a singlet of, say, SU (2) L and highest weight of SU (2) R in the SO (4) ⊂ SU (4). Furthermore, the two signs in (3.5) correspond to the twochiralities of the SO (4) isometry group of S . Since this S corresponds to the S inside AdS , this checks with the above requirement of b = ± χ and ˜ χ ( b ) factorise in each component of thegravitino variation equation which then becomes equivalent to the following system ofcoupled differential and algebraic equations on ε (cid:20) ˜ ∇ µ − F µν Ξ νm γ m γ ˆ σ s + i A µ s − (cid:18) ˜ G/ + ˜ V/γ ˆ σ s + i˜ gs (cid:19) γ ˆ σ γ µ (cid:21) ε = 0 (3.7) (cid:20) i2 ρ ρ γ ˆ σ + 12 ∂/ρ + ρ (cid:18) ˜ G/ + ˜ V/γ ˆ σ s − i˜ gs (cid:19) γ ˆ σ (cid:21) ε = 0 (3.8) (cid:20) i2 (cid:18) − ρ ρ (cid:19) γ ˆ σ + 12 ∂/ρ + 18 ρ F/γ ˆ σ s + ρ (cid:18) ˜ G/ − ˜ V/γ ˆ σ s + i˜ gs (cid:19) γ ˆ σ (cid:21) ε = 0 (3.9) (cid:20) i2 bγ ˆ σ + 12 ∂/ ˜ ρ − ˜ ρ (cid:18) ˜ G/ + ˜ V/γ ˆ σ s + i˜ gs (cid:19) γ ˆ σ (cid:21) ε = 0 . (3.10)Note that the first equation is a first order differential 4-vector equation for ε while thelast three are algebraic 4-scalar equations. For example the first equation is obtained as follows( ∇ µ + M Γ µ ) ψ = (cid:18) ˜ ∇ µ − ρ F µν Ξ νm Γ m Γ ˆ3 + A µ (cid:16) Σ ˆ3 + Γ ˆ1 Γ ˆ2 (cid:17) − A µ ∇ ˆ3 + M Γ µ (cid:19) ψ == (cid:18) ˜ ∇ µ − ρ F µν Ξ νm Γ m Γ ˆ3 + A µ Γ ˆ1 Γ ˆ2 + M (Γ µ + A µ ρ Γ ˆ3 ) (cid:19) ψ == (cid:18) ˜ ∇ µ − ρ F µν Ξ νm γ m σ ˆ3 + A µ σ ˆ3 + M γ µ (cid:19) ψ (3.6)
7e now express all the supergravity fields via the functions m, n, p and T of the previoussection. We are thus guaranteed that a solution to the above system with b = s = 1 existsby the analysis in [12]. We now ask that a second solution to these equations exists for b = − s = 1.We have used Mathematica to solve explicitly the equations. The existence of solutionsimplies certain constraints on the background, which are more conveniently expressed interms of the metric entries as1 + A t = ρ /ρ ρ − ρ = S /ρ T ∂ y ln( ρ /ρ ) (cid:2) ρ /ρ − y∂ y ln( ρ /ρ ) (cid:3) + y (cid:2) ∂ r ln( ρ /ρ ) (cid:3) = 0 (3.11)where ( r, φ ) are polar coordinates in the ( x , x ) plane. Notice that the first two constraintstogether with the relation y = ρ ˜ ρ allow us to express the four functions ρ , ρ , ˜ ρ, A t interms of only one function. The last constraint together with the equation for ∂ y T canbe used to eliminate T . Moreover, the three second order differential equations that camefrom the integrability condition for the 1 / z ˜ z ≡ (cid:20) (cid:18) ρ (1 + A t )˜ ρ (cid:19)(cid:21) (3.12)1 r ∂ r (cid:0) r∂r ˜ z (cid:1) + y∂ y (cid:26) T y (cid:20) ∂ y ˜ z + 4˜ z (1 − ˜ z ) ρ /ρ − y (cid:21)(cid:27) = 0 (3.13)where the combination ρ /ρ is given by ρ ρ = q − S − ˜ zy ˜ z − (3.14)and T can be found by solving the third equation in (3.11). The solution is thus specifiedcompletely by a single function . AdS × Y p,q metrics In this section, we are going to show how the
AdS × Y p,q geometries arise from the genericdescription given above. As a first step we present the conditions that should be satisfiedby the 1 / AdS × X , (3.15) If, instead of doubling supersymmetry by requiring b = − s = 1 in addition to b = s = 1, onerequires to have solutions of (3.1) also for b = s = −
1, then one obtains a different set of constraints onthe background. By making an asymptotic analysis similar to the one we will perform here in Section 4,it can be shown that the resulting geometry describes 1/4-BPS states in the background
AdS × S X . These will turn out to be equivalent to a singlefirst order differential equation which implies the second order equation in (3.13). Theopposite in general cannot be proven: the generic solution preserving 8 supersymmetriesapparently is not factorisable in general.As a second step we will prove, by giving the explicit coordinate transformation to thegauge in [16], that the X factor is indeed a generic Y p,q manifold.First of all we switch to the more convenient coordinates ( ˜ ρ, ρ , ˜ φ, ˆ ψ ′ ) defined by y = ρ ˜ ρr = r ( ρ , ˜ ρ ) φ = ˜ φ + ˜ c t ˆ ψ = ˆ ψ ′ − γ t − δ φ ≡ ˆ ψ ′ − (2 γ + 2˜ c δ ) t − δ ˜ φ (3.16)Using the constraints in (3.11) the solution is completely specified once the explicit formof the function r ( ρ , ˜ ρ ) is known.The last shift implies that the left invariant one-form σ ˆ3 is shifted to σ ˆ3 ′ + ( γ + ˜ c δ )d t + δ d ˜ φ . With a slight abuse of notation we will keep calling this shifted one form σ ˆ3 . Themetric of 2.1 is thusd s = − h − (d t + V φ d φ ) + h ρ ρ ( T δ ij d x i d x j + d y )++ ˜ ρ dΩ + ρ (cid:2)(cid:0) σ ˆ1 (cid:1) + (cid:0) σ ˆ2 (cid:1) (cid:3) + ρ ( σ ˆ3 − A t d t − A φ d φ ) == g tt d t + g ˜ ρ ˜ ρ d ˜ ρ + ˜ ρ d ˜Ω + 2 g t ˜ φ d t d ˜ φ + 2 g ρ ˜ ρ d ρ d ˜ ρ ++ g ρ ρ d ρ + g ˜ φ ˜ φ d ˜ φ + ρ (cid:2)(cid:0) σ ˆ1 (cid:1) + (cid:0) σ ˆ2 (cid:1) (cid:3) ++ ρ (cid:20) σ ˆ3 + (cid:0) γ − A t − ˜ c ( A φ − δ ) (cid:1) d t − ( A φ − δ )d ˜ φ (cid:21) , (3.17)with g tt = − h − (1 + ˜ cV φ ) + ˜ c h ρ ρ T r g ˜ ρ ˜ ρ = h ρ ρ " ρ + T r (cid:18) ∂ ln r∂ ˜ ρ (cid:19) g t ˜ φ = − h − (1 + ˜ cV φ ) V φ + ˜ ch ρ ρ T r g ρ ˜ ρ = h ρ ρ (cid:20) ρ ˜ ρ + T r ∂ ln r∂ ˜ ρ ∂ ln r∂ρ (cid:21) g ρ ρ = h ρ ρ " ˜ ρ + T r (cid:18) ∂ ln r∂ρ (cid:19) g ˜ φ ˜ φ = − h − V φ + h ρ ρ T r (3.18)9e recall the constraint on the metric components coming from the requirement of 1 / A t = ρ ρ ρ − ρ = S ρ h − = ˜ ρ + ρ /ρ . (3.19)In order that the metric factorises we need the d t σ ˆ3 term to vanish which requires that A t + ˜ c ( A φ − δ ) = γ . (3.20)Imposing also g t ˜ φ = 0 we obtain˜ ch ρ ρ T r = h − (1 + ˜ cV φ ) V φ . (3.21)In order to have an AdS factor we should have − g tt = L + ˜ ρ which gives, using the lastrelation h − (1 + ˜ cV φ ) = L + ˜ ρ , (3.22)We also demand that g ˜ ρ ˜ ρ = L L +˜ ρ which after a little bit of algebra gives ∂ ln r∂ ˜ ρ = ± ˜ c ˜ ρL + ˜ ρ . (3.23)Requiring that we have a product metric means that we also must impose that g ρ ˜ ρ = 0which implies ∂ ln r∂ρ = ∓ ˜ c ρ ρ L − ρ . (3.24)As a result we find immediately that g ρ ρ = ρ L ρ L − ρ (3.25)and g ˜ φ ˜ φ = 1˜ c (cid:18) L − ρ ρ (cid:19) . (3.26)The generic solutions to equation (3.23) for the upper sign are r = ( L + ˜ ρ ) ˜ c/ r ( ρ ) ρ ˜ c (3.27)where we have extracted the ρ ˜ c for future convenience. (3.24) is an equation for r ( ρ ) r ′ ( ρ ) = ˜ c L ( ρ − S ) ρ − L ρ ( ρ − S ) r ( ρ ) (3.28)10sing the last constraint in (3.11) T ∂ y ln( ρ /ρ ) (cid:2) ρ /ρ − y∂ y ln( ρ /ρ ) (cid:3) + y (cid:2) ∂ r ln( ρ /ρ ) (cid:3) = 0 (3.29)we can find T . Note that both the first order differential equation for T∂ y ln T = D (3.30)and the second order equation in (3.13) are satisfied when r ( ρ ) satisfies the equation(3.28). Y p,q coordinates Now we show the coordinates transformation that brings the metric on X to the standardmetric on Y p,q as presented in [16]. We perform the rescaling˜ ρ → L ˜ ρ, ρ i → Lρ i , S → LS (3.31)which takes the metric of AdS into the formd s AdS = L (cid:18) − ( ˜ ρ + 1)d t + d ˜ ρ ˜ ρ + 1 + ˜ ρ dΩ (cid:19) (3.32)while the metric on the “internal” part isd s = L (cid:20) ρ ρ − ρ d ρ + 1˜ c (cid:18) − ρ ρ (cid:19) d ˜ φ ++ ρ (cid:2)(cid:0) σ ˆ1 (cid:1) + (cid:0) σ ˆ2 (cid:1) (cid:3) + ρ (cid:0) σ ˆ3 − ( A φ − δ )d ˜ φ (cid:1) (cid:21) . (3.33)The standard form for the metric on Y p,q [16] is,d s = 1 − c ˆ y θ + sin ˆ θ d ˆ φ ) + 1 w (ˆ y ) q (ˆ y ) d ˆ y + q (ˆ y )9 (d ˆ ψ + cos ˆ θ d ˆ φ ) + w (ˆ y ) (cid:20) d α − ac − y + ˆ y c a − ˆ y ) (d ˆ ψ + cos ˆ θ d ˆ φ ) (cid:21) (3.34)with w (ˆ y ) = 2( a − ˆ y )1 − c ˆ yq (ˆ y ) = a − y + 2 c ˆ y a − ˆ y . (3.35) Notice that the ˆ ψ of [16] has the opposite sign to that used here. σ ˆ1 = − (cos ˆ ψ d ˆ θ + sin ˆ ψ sin ˆ θ d ˆ φ ) σ ˆ2 = − ( − sin ˆ ψ d ˆ θ + cos ˆ ψ sin ˆ θ d ˆ φ ) σ ˆ3 = − ( d ˆ ψ + cos ˆ θ d ˆ φ ) (3.36)we immediately get ρ = 23 (1 − c ˆ y ) (3.37)and 14 ρ = 2 + ac − c ˆ y + 3 c ˆ y − c ˆ y ) . (3.38)Recalling that ρ − ρ = S /ρ we have S = 427 (1 − ac ) . (3.39)We also have A φ = 1˜ c ( γ − A t ) + δ = 1˜ c (cid:18) γ + 1 − ρ ρ (cid:19) + δ . (3.40)Assuming that α = β ˜ φ and equating the d ˜ φ component of the metric we get1˜ c (cid:18) − ρ ρ (cid:19) + ρ ( A φ − δ ) = w (ˆ y ) β (3.41)which implies after some straightforward algebra that γ = 12 , β = ± c c . (3.42)The coefficient of the cross term d ˜ φ σ ˆ3 is12 ρ ( A φ − δ ) = − βw (ˆ y ) ac − y + ˆ y c a − ˆ y ) (3.43)which implies that we must have β = − c c . We can therefore set β = − c = 12 c . (3.44)Using the expression for r r = ( L + ˜ ρ ) ˜ c/ r ( ρ ) ρ ˜ c r ′ ( ρ ) = ˜ c L ( ρ − S ) ρ − L ρ ( ρ − S ) r ( ρ ) (3.45)and the definition A φ = A t V φ + 12 r∂ r ln T (3.46)12e get A t + ˜ c ( A φ − δ ) = 12 − ˜ c δ ) (3.47)which gives δ = −
12 (3.48)for c, ˜ c = 0. Finally, the matching of the dˆ y factor19 c ρ − ρ = 1 w (ˆ y ) q (ˆ y ) (3.49)is identically satisfied. Finally, we observe that the polynomial q (ˆ y ) = a − y + 2 c ˆ y ,whose zeroes ˆ y and ˆ y , with ˆ y < y the smallest between the two other positivezeroes, determine the range of ˆ y , ˆ y ≤ ˆ y ≤ ˆ y , can be expressed in terms of ρ as q = − ρ − ρ + S ) /
4. Notice also that in the metric (3.34) any non zero value of c can bereabsorbed in a rescaling of ˆ y and α . We may thus set c = 1 whenever c = 0. c=0 case Let us now take a look at the singular case c = ˜ c = 0 (3.50)which corresponds to the Sasaki-Einstein internal manifold Y , ≡ T , . From the equations(3.21),(3.47) we can immediately obtain A t = γ = 12 V φ = 0 (3.51)and from (3.37),(3.38),(3.39) ρ = ρ = 23 S = 427 . (3.52)Notice that the equation for ρ implies y = r L ˜ ρ (3.53)Given these explicit values for ρ and ρ , the last constraint in (3.11) T ∂ y ln( ρ /ρ ) (cid:2) ρ /ρ − y∂ y ln( ρ /ρ ) (cid:3) + y (cid:2) ∂ r ln( ρ /ρ ) (cid:3) = 0 (3.54)13s automatically satisfied.The metric on the internal manifold becomesd s = L (cid:20) τ ( r ) ( d r r + d ˜ φ ) + 23 (cid:2)(cid:0) σ ˆ1 (cid:1) + (cid:0) σ ˆ2 (cid:1) (cid:3) + 49 (cid:0) σ ˆ3 − ( A φ − δ )d ˜ φ (cid:1) (cid:21) (3.55)where T = ( ˜ ρ + 1) τ ( r ) r (3.56)is such that T solves the equation ∂ y ln T = D ⇐⇒ ∂ ˜ ρ ln T = 2 ˜ ρ ˜ ρ + 1 (3.57)We now match this expression with the one in [16]. For c = 0, a can be reabsorbed ina coordinate redefinition. We set, for convenience, a = 3 (3.58)and obtain,d s = 16 (dˆ θ + sin ˆ θ d ˆ φ ) + 16(1 − ˆ y ) dˆ y + 1 − ˆ y − ˆ y ) (d ˆ ψ + cos ˆ θ d ˆ φ ) + 2(3 − ˆ y ) (cid:20) d α + 2ˆ y − ˆ y ) (d ˆ ψ + cos ˆ θ d ˆ φ ) (cid:21) (3.59)Assuming, as in the generic case, α ≡ − ˜ φ , and equating the g α and g αα components weget A φ − δ = − y (3.60)and 32 τ + 4ˆ y = 2(3 − ˆ y ) ⇒ τ = 4(1 − ˆ y ) (3.61)Assuming r = r (ˆ y ) and equating the dˆ y term gives ∂ ln r∂ ˆ y = ± − ˆ y ) ⇒ r = λ (cid:18) − ˆ y y (cid:19) ∓ / (3.62)where λ is an arbitrary constant and we fix λ = 1. We are now able to determine theconstant δ through the equation A φ = A t V φ + 12 r∂ r ln T = − ∓ y (3.63)which fixes the upper choice for the sign and δ = −
12 (3.64)14n order to bring the metric to the standard T , form we setˆ y = − cos ˜ θ (3.65)which gives r = tan ˜ θ ! / , τ = 4 sin ˜ θ (3.66)and thusd s L = 16 (cid:0) d˜ θ + 36 sin ˜ θ d ˜ φ (cid:1) + 16 (cid:0) dˆ θ + sin ˆ θ d ˆ φ (cid:1) + 19 (cid:0) d ˆ ψ + cos ˆ θ d ˆ φ + 6 cos ˜ θ d ˜ φ (cid:1) (3.67)which is the T , metric up to the trivial rescaling˜ φ →
16 ˜ φ (3.68) AdS × Y p,q In this section we study generic asymptotic perturbations of the
AdS × Y p,q geometries thatpreserve 1 / AdS × Y p,q and solve the differential equations (2.26) with the boundary conditionsthat the solutions approach AdS × Y p,q at large distances (including also the particularcase c = 0). We will work in the somewhat mixed coordinates ( y, ˆ y ) or ( y, ˜ θ ) and solve theequation in an expansion for large y , with the simplifying assumption that the solutionsare invariant under shifts in ˜ φ . We make the following Ansatz for the expansion of ourfunctions, ρ = L r
23 (1 − c ˆ y ) (cid:18) ρ (1)1 (ˆ y ) L y + ρ (2)1 (ˆ y ) L y (cid:19) (4.1) ρ = L s ac − c ˆ y + 3 c ˆ y )9(1 − c ˆ y ) (cid:18) ρ (1)3 (ˆ y ) L y + ρ (2)3 (ˆ y ) L y (cid:19) (4.2)˜ ρ = yρ (4.3) A t = 1 − ac ac − c ˆ y + 3 c ˆ y (cid:18) A (1) t (ˆ y ) L y + A (2) t (ˆ y ) L y (cid:19) (4.4) T = yr s a − y + 2 c ˆ y )(1 − c ˆ y ) (cid:18) t (1) (ˆ y ) L y + t (2) (ˆ y ) L y (cid:19) (4.5) V φ = 4 c (1 − c ˆ y )( a − y + 2 c ˆ y )3(2 + ac − c ˆ y + 3 c ˆ y ) L y + V (2) φ (ˆ y ) L y + V (3) φ (ˆ y ) L y (4.6) r = y c/ r (ˆ y ) (4.7)15his expansion reproduces the c = 0 limit upon setting a = 3, as in the previous section.In these coordinates, the condition (3.28) becomes r ′ (ˆ y ) = 2 + ac − c ˆ y + 3 c ˆ y − c ˆ y )( a − y + 2 c ˆ y ) r (ˆ y ) (4.8)The functions m, n, p are given by m = 1 ρ [ ˜ ρ + (1 + A t ) ρ ] (4.9) n = (1 + A t ) ρ y [ ˜ ρ + (1 + A t ) ρ ] (4.10) p = ρ y [ ˜ ρ + (1 + A t ) ρ ] . (4.11)The constraints in (3.11) and the equations (3.12) are satisfied at leading order in y . Werewrite the generic equations (2.26) in polar coordinates dividing them by T and exploitingthe U (1) symmetry of our solutions y r T r∂ r ( r∂ r n ) + ∂ y (cid:0) y ∂ y n (cid:1) + y ∂ y (cid:2)(cid:0) yDn + 2 y m ( n − p ) (cid:1)(cid:3) + 2 y D (2 n + y∂ y n + yDn ) = 0 y r T r∂ r ( r∂ r m ) + ∂ y (cid:0) y ∂ y m (cid:1) + ∂ y (cid:0) y mD (cid:1) + 2 Dy ( ∂ y m + 2 Dm ) = 0 y r T r∂ r ( r∂ r p ) + ∂ y (cid:0) y ∂ y p (cid:1) + ∂ y (cid:2) y ny ( n − p ) (cid:3) + 2 Dy [ ∂ y p + 4 ny ( n − p )] = 0 ∂ y ln T = D . (4.12)where ∂ y f ( y, ˆ y ) ≡ d f d y (cid:12)(cid:12)(cid:12)(cid:12) r = − c r (ˆ y )2 r ′ (ˆ y ) d f dˆ y (cid:12)(cid:12)(cid:12)(cid:12) y + d f d y (cid:12)(cid:12)(cid:12)(cid:12) ˆ y (4.13) r∂ r f ( y, ˆ y ) ≡ r d f d r (cid:12)(cid:12)(cid:12)(cid:12) y = r (ˆ y ) r ′ (ˆ y ) d f dˆ y (cid:12)(cid:12)(cid:12)(cid:12) y (4.14)The generic asymptotic solutions to these equation are specified, at each order, by 7integration constants. As in [12], requiring regularity of the solutions implies that not allof them are independent and indeed we have only three independent integration constants.For the case of T , asymptotics, specified by c = 0 the first subleading corrections are16iven by: ρ (1)1 (˜ θ ) = − k + C cos ˜ θ (4.15) ρ (1)3 (˜ θ ) = ρ (1)1 (˜ θ ) + k (1) (˜ θ ) (4.16) k (1) (˜ θ ) = k (4.17) A (1) t (˜ θ ) = C − C cos ˜ θ (4.18) t (1) (˜ θ ) = L p / k ) sin ˜ θ tan ˜ θ (4.19) V (2) φ (˜ θ ) = − C sin ˜ θ (4.20)while in the generic case we get: ρ (1)1 (ˆ y ) = A [2 c K + 9 Ak + 4ˆ yBC ]6(2 + ac − c ˆ y + 3 c ˆ y ) ρ (1)3 (ˆ y ) = ρ (1)1 (ˆ y ) + k (1) (ˆ y ) k (1) (ˆ y ) = A [4 c LK + 9 (cid:0) − c ˆ y − c ˆ y − ac (4 − c ˆ y ) (cid:1) k ]6(2 + ac − c ˆ y + 3 c ˆ y ) A (1) t (ˆ y ) = (cid:0) − c A K (2 + ac − c ˆ y + 3 c ˆ y ) + 2 AL C − AC + − c A [ a c + ˆ y (12 − c ˆ y + 21 c ˆ y − c ˆ y ) + 2 a ( − c ˆ y − c ˆ y + c ˆ y )]2 LB kt (1) (ˆ y ) = A (4 L − k )6 B where A = 1 − c ˆ yB = 2 + ac − c ˆ y + 2 c ˆ y K = a − y + 2 c ˆ y L = 1 − ac The three arbitrary integration constants, C , C , k will turn out to be related to thesupergravity dual of the flavour and baryon charge of the solutions. The second orderregular solutions are rather complicated. In general, they will involve new integrationsconstants together with a inhomogeneous part. The expressions for the inhomogeneouspart can be found in the Appendix.As already noticed, any c = 0 can be reabsorbed by a redefinition of ˆ y and so we set c = 1. 17 U (1) charges We will now show how the first subleading corrections described in the previous section giverise to the Kaluza-Klein reduction of type IIB supergravity on the Y p,q manifolds respectingthe symmetry of our Ansatz. We will calculate the global charges of the solutions underthree U (1) massless KK gauge fields living in AdS ; two of them can be identified withthe KK modes of the metric associated to the Killing vectors ∂ α and ∂ ˆ ψ and which aredual to the flavour and R charges of the dual quiver gauge theory (more precisely to linearcombinations of the charges) while the third one is associated to the expansion of the RR4-form potential on the cohomology of Y p,q and it is dual to the baryon charge of the gaugetheory. Since the third Betti number of such manifolds is one there is only one baryoncharge.In general, the metric on the compact manifold is modified by the metric KK gaugefields as d s = g αβ (d ξ α + K αI A Iµ d x µ )(d ξ β + K βI A Iµ d x µ ) (5.1)where ξ α are coordinates in Y p,q and x µ in AdS and K I = K αI ∂ α I = 1 , . . . , n (5.2)are n Killing vectors of Y p,q . In our case, only two gauge fields are turned on and they areassociated to ∂ α and ∂ ˆ ψ . We denote the two global gauge charges respectively as J and Q .The leading order of the corresponding gauge fields A J , A Q is thus given by A J = J ˜ ρ d t A Q = Q ˜ ρ d t . (5.3)The metric is modified by the shiftsd ˆ ψ → d ˆ ψ + Q ˜ ρ d t (5.4)d α → d α + J ˜ ρ d t (5.5)tod s L − = 1 − c ˆ y θ + sin ˆ θ d ˆ φ ) + 1 w (ˆ y ) q (ˆ y ) d ˆ y + q (ˆ y )9 (d ˆ ψ + Q ˜ ρ d t + cos ˆ θ d ˆ φ ) + w (ˆ y ) (cid:20) d α + J ˜ ρ d t − ac − y + ˆ y c a − ˆ y ) (d ˆ ψ + Q ˜ ρ d t + cos ˆ θ d ˆ φ ) (cid:21) . (5.6)Given the expression above for the metric and the solution of the equations of motionup to the first sub-leading order we obtain Q = 3 C − C , (5.7) J = 12 C − C . (5.8)18imilarly, in the case of T , we haved s L − = 16 (cid:0) d˜ θ + 36 sin ˜ θ (d ˜ φ + J ˜ ρ d t ) (cid:1) + 16 (cid:0) dˆ θ + sin ˆ θ d ˆ φ (cid:1) ++ 19 (cid:0) d ˆ ψ + Q ˜ ρ d t + cos ˆ θ d ˆ φ + 6 cos ˜ θ (d ˜ φ + J ˜ ρ d t ) (cid:1) (5.9)with Q = 32 C (5.10) J = − C (5.11) R -charge and Reeb vector In order to correctly identify the R charge we proceed as in [27, 16, 28]. We define the newcoordinates ˆ ψ ′ = ˆ ψ (5.12) β = − α + c ˆ ψ (5.13)In this coordinate system we can write the metric as a local U (1) fiber over a Kaehler-Einstein manifold and ˆ ψ ′ is a coordinate on the local U (1) fiber. From (2.17) of [16], wehave dΩ Y p,q = ( e ˆ θ ) + ( e ˆ φ ) + ( e ˆ y ) + ( e β ) + ( e ˆ ψ ) (5.14)where the one forms on Y p,q are, e ˆ θ = r − c ˆ y d ˆ θ , e ˆ φ = r − c ˆ y θ d ˆ φ , (5.15) e ˆ y = 1 √ wq dˆ y , e β = √ wq β + c cos ˆ θ d ˆ φ ) , (5.16) e ˆ ψ ′ = 13 ( − d ˆ ψ ′ − cos ˆ θ d ˆ φ + ˆ y (d β + c cos ˆ θ d ˆ φ )) . (5.17)As noted in [27], the R -symmetry is identified with a shift in the angular variable ψ R = −
12 ˆ ψ ′ (5.18)at constant β . As a consequence, the U (1) R -symmetry gauge field is given by A R = − A Q (5.19)19nd Q R = − Q . (5.20)The associated Killing vector is given by K R = − ∂ ˆ ψ − c ∂ α (5.21)which coincides with the Reeb vector of the Sasaki-Einstein manifold. Notice that ourReeb vector differs by a factor of 2/3 from the one in [15],[16]. The self-dual Ramond-Ramond field strength F (5) can be written as F (5) = F + ⋆ F . (6.1)With our conventions and normalisations, the leading order for F is given by F = L V ol ( Y p,q ) (6.2)where V ol ( Y p,q ) is the volume form of the unit radius Y p,q . The background metric isperturbed by the KK gauge fields as described in the previous chapter: the field strengthis also perturbed in order to satisfy the equations of motion. The corrections are knownto be of the form [27, 28, 29] F = L d ( A Q ∧ ω Q + A J ∧ ω J + A B ∧ ω B ) . (6.3)The Y p,q three forms ω I are defined byd ω I + ι K I V ol ( Y p,q ) = 0 (6.4)where K I , I = J, Q is the Killing vector of Y p,q associated with the A I gauge field. The3-forms ω J,Q are clearly defined up to the addition of a closed form. The 3-form ω B is thegenerator of the one dimensional cohomology of the Sasaki-Einstein manifold and A B isthe gauge field dual to the baryon current of the CFT. The arbitrary shift by a closed formof the ω J,K corresponds to the possibility of shifting the mesonic symmetries of the theoryby an arbitrary baryonic one.In the case of generic Y p,q for c = 0 we obtain the following form for the subleadingcorrections to F (5) , F = − ρ d ˜ ρ ∧ d t ∧ (cid:26) k (cid:28)(cid:16) σ ˆ3 − y (1 − ˆ y )d α (cid:17) σ ˆ1 ∧ σ ˆ2 − − ˆ y ) σ ˆ3 ∧ d α ∧ dˆ y (cid:21) ++ Q (cid:20)(cid:18) a − σ ˆ3 − a − y ( a − − y + 2ˆ y − ˆ y ) d α (cid:19) σ ˆ1 ∧ σ ˆ2 + 2 + a − y + 3ˆ y − ˆ y ) σ ˆ3 ∧ d α ∧ dˆ y (cid:21) + J (cid:20)(cid:18) a − y + ˆ y σ ˆ3 − a − a ˆ y + ˆ y − ˆ y ) d α (cid:19) σ ˆ1 ∧ σ ˆ2 − a − y + ˆ y − ˆ y ) σ ˆ3 ∧ d α ∧ dˆ y (cid:21) (cid:27) ++ 1˜ ρ d t ∧ (cid:18) − Q − ˆ y )9 d α − J − ˆ y )9 σ ˆ3 (cid:19) ∧ dˆ y ∧ σ ˆ1 ∧ σ ˆ2 (6.5)20hile for the case c = 0 and going to the natural coordinate (˜ θ, ˜ φ ) defined by (ˆ y, α ) =( − cos ˜ θ, − ˜ φ ) we get F = − ρ d ˜ ρ ∧ d t (cid:26) − k h(cid:16) σ ˆ3 − θ d˜ θ ∧ d ˜ φ (cid:17) σ ˆ1 ∧ σ ˆ2 − θσ ˆ3 ∧ d˜ θ ∧ d ˜ φ i + Q (cid:20)(cid:18) σ ˆ3 − cos ˜ θ d ˜ φ (cid:19) σ ˆ1 ∧ σ ˆ2 + 12 sin ˜ θσ ˆ3 ∧ d˜ θ ∧ d ˜ φ (cid:21) + J (cid:20)(cid:18) −
43 cos ˜ θσ ˆ3 + 12 (1 + 7 cos 2˜ θ )d ˜ φ (cid:19) σ ˆ1 ∧ σ ˆ2 + 12 sin 2˜ θσ ˆ3 ∧ d˜ θ ∧ d ˜ φ (cid:21) (cid:27) + 1˜ ρ d t ∧ (cid:18) Q
29 d ˜ φ + J σ ˆ3 (cid:19) ∧ sin ˜ θ d˜ θ ∧ σ ˆ1 ∧ σ ˆ2 (6.6)The volume form on Y p,q is given by Vol( Y p,q ) = − e ˆ y ∧ e β ∧ e ˆ θ ∧ e ˆ φ ∧ e ˆ ψ ′ = 4(1 − c ˆ y )9 dˆ y ∧ d α ∧ σ ˆ1 ∧ σ ˆ2 ∧ σ ˆ3 , (6.7)and we define the three forms ω ± ≡ e ˆ ψ ′ ∧ ( e ˆ θ ∧ e ˆ φ ± e ˆ y ∧ e β ) == 13 (cid:16) σ ˆ3 (1 − c ˆ y ) − y d α (cid:17) ∧ (cid:18) − c ˆ y )3 σ ˆ1 ∧ σ ˆ2 ∓
13 dˆ y ∧ ( cσ ˆ3 + 3d α ) (cid:19) (6.8)The local K¨ahler form J is given by J = e ˆ θ ∧ e ˆ φ − e ˆ y ∧ e β = 12 d e ˆ ψ ′ (6.9)The closed form ω B is given as in [28] by ω B = 98 π (1 − c ˆ y ) ( p − q ) ω − (6.10)With this normalisation and assuming that A B = Q B ˜ ρ d t , the baryon charge Q B is given by Q B = π p − q ) k . (6.11)In the case of T , and recalling the change of coordinates (ˆ y, α ) = ( − cos ˜ θ, − ˜ φ ) we get ω ± = (cid:18) σ ˆ3 − θ d ˜ φ (cid:19) ∧ (cid:18) σ ˆ1 ∧ σ ˆ2 ± sin ˜ θ d˜ θ ∧ d ˜ φ (cid:19) (6.12)with ω B ≡ π ω − (6.13) The orientation is chosen to satisfy (6.2) Q B = 2 π k . (6.14)We now rewrite the expansion of F as F = L d ( A R ∧ ω R + A β ∧ ω β + A B ∧ ω B ) . (6.15)where A R = − A Q , A β = − A J − A Q (6.16)are the gauge fields associated to the Killing vectors K R = − ∂ ˆ ψ − ∂ α , ∂ β = − ∂ α . (6.17)The remaining 3-forms are given by ω R = − ω + (6.18) ω β = − ( a − y + ˆ y )18 σ ˆ3 ∧ (cid:18) σ ˆ1 ∧ σ ˆ2 − − ˆ y ) d α ∧ dˆ y (cid:19) + (6.19) − a − a ˆ y + ˆ y − ˆ y ) d α ∧ σ ˆ1 ∧ σ ˆ2 . (6.20)It can be shown without difficulty that they satisfy the expected relationsd ω R + ι ( − ∂ ˆ ψ − ∂ α ) V ol ( Y p,q ) = 0 , (6.21)d ω J + ι ∂ β V ol ( Y p,q ) = 0 . (6.22) AdS × X AdS × Y p,q spacetimes under examination. There has been a considerable amount of work over theyears on the definition of mass and other conserved charges in general relativity. Theissue becomes even subtler in the case of the definition of the mass in asymptotically AdS spaces. For example, the standard expression given in terms of a Komar integral gives adivergent result in this case, and the procedure of renormalisation is ambiguous. We willfollow the definition of conserved charges given by Wald and collaborators [26, 25] whichprovides a possible general framework for addressing this issue, and apply it to our casefor the computation of the mass. Since our solutions mix beyond the leading order
AdS and Y p,q coordinates, it is natural to take a ten dimensional approach for the definitionof mass, which has the advantage of being relatively simple both from the conceptual andfrom the technical point of view.Another derivation of conserved charges applicable to spacetime with AdS asymptotics22more precisely asymptotically locally AdS spacetimes) was presented in [30]. Using nonlinear KK mapping one can also uplift this derivation to ten dimensional asymptotically
AdS × X backgrounds [31, 32].The main result of this section is to prove that, with the adopted definition of mass,the expected BPS relation: M L = 32 R (7.1)which is a consequence of the N = 1 superconformal algebra on the field theory side, issatisfied. AdS × X We are dealing with an asymptotically
AdS × X spacetime, where X is a compactmanifold.It is convenient to choose coordinates such that, defining a radial AdS coordinate Ω, g ΩΩ = L / Ω and g Ω M = 0 for M = Ω, M denoting a ten dimensional coordinate. We willalso denote the AdS coordinates with µ, ν, . . . and the internal coordinates with a, b, . . . .At leading order for large Ω we haved s = L Ω (cid:2) dΩ − d t + dΩ (cid:3) + L d s ( Y p,q ) . (7.2)We will keep corrections to orders Ω k , with k = 0 , , AdS part, g µν , k = 1 , g ab , and mixed parts, g µa respectively. There are of course corrections ofhigher order in Ω to the background 5-form given by the volume forms on AdS and Y p,q which we will discuss later.In general the construction of conserved charges proceeds as follows: let us denote forthe moment as ϕ the generic field appearing in a Lagrangian L . The variation of theLagrangian with respect to ϕ is given by δ L = E ( ϕ ) δϕ + d θ ( ϕ, δϕ ) . (7.3)where E ( ϕ ) denotes the equations of motion. This defines the symplectic potential θ ,corresponding to the boundary term that arises from integrating by parts in order toremove derivative of δϕ . It is a 9-form in spacetime.We will be interested in the following asymptotic symmetry generator ξ = ∂∂t (7.4)We want to identify the Hamiltonian generator H ξ of such symmetry. Its value on thedesired solution will be our definition of the mass of the metric H ξ is defined via its We are specifying here to a particular symmetry generator since we are interested in the mass, butthe same procedure con be applied to the most general asymptotic symmetry generator [26]. δφ , obeying the linearised equations ofmotion in a given background obeying the full equations of motion [26]: δ H ξ = Z ∂ Σ ( δQ ξ − ξ · θ ) (7.5)where Σ is a 9 dimensional submanifold of the spacetime without boundary, a “slice”corresponding to the vector field ξ . By the integral over ∂ Σ we mean a limiting process inwhich the integral is first taken over the boundary ∂K of a compact region inside Σ andthen K approaches Σ in a suitable manner. The 8-form Q ξ is the Noether charge of theasymptotic symmetry ξ . It has a contribution coming from the gravitational lagrangian: Q gravα ··· α = − πG ∇ b ξ c ǫ bca ··· a . (7.6)where ǫ = √− det g d x is the volume form. Also, the gravitational contribution to θ is: θ grava ··· a = 116 πG v a ǫ aa ··· a (7.7)with v a = ∇ b δg ab − ∇ a δg bb (7.8)Finally, the RR 5-form contributes both to Q ξ and θ , giving rise to a single term in thecombination δQ x i − ξ · θ . With our normalisation for the 5-form F , the final result for δ H x i is δ H ξ = Z ∂ Σ πG (cid:0) − δQ gravξ − ξ a ( v a ǫ aa ··· a − F a ··· a δA a ··· a ) (cid:1) (7.9)where F (5) = d A (4) .Under mild assumptions [26], a necessary and sufficient condition for the existence of H ξ is the integrability of the equation for H ξ :( δ δ − δ δ ) H ξ = 0 (7.10) i. e. ξ · [ δ θ ( ϕ, δ ϕ ) − δ θ ( ϕ, δ ϕ )] (7.11)When this condition is satisfied we are guaranteed that an 8-form I ξ exists whose variationis δI ξ = δQ ξ − ξ · θ (7.12)The value of the global charge associated with the asymptotic isometry generated by ξ isgiven by a simple “surface” integral, up to an arbitrary constant which can be determinedby fixing the value of the charge for a “reference solution”. H ξ = Z ∂ Σ I ξ + H ξ . (7.13)24otice that the 8-manifold ∂ Σ in the present case reduces asymptotically to S × Y pq , where S is a 3-sphere of radius L/ Ω inside
AdS . The existence of H can be explicitly checkedfor a background with the asymptotic behaviour we have discussed above for the metric.The expression for θ grav in our gauge is proportional to ξ · θ grav ( δg ) = (cid:0) Ω δ ( g tM ∂ Ω g aM √ g ) − g MN √ g ( ∂ Ω δg MN − Γ P Ω M δg P N ) (cid:1) ǫ t Ω M ...M (7.14)One can verify, using the asymptotic expansion for the metric given before, that δ [1 θ ( δ g ) =0. The crucial fact for this result to hold is that g MN δg MN = O (Ω ). This is satisfied byour BPS solutions, but can be proven to hold more generally, even for non necessarily BPSsolutions of the equations of motion, given an appropriate asymptotic behaviour [33]. Onecan similarly verify that the contribution of the 5-form to θ is integrable.Once we have verified the existence of H ξ , we can define the mass of a generic solution M to the equations of motion as the value of H ξ on such a solutions M M ≡ H ξ | M . (7.15) R -charge We will now proceed to the calculation of the mass and R -charge for the solutions we havedescribed in the previous Sections. We are interested in the dependence of the mass M =on the integration constants, C , C and k . Therefore we will compute, ∂M∂C i and ∂M∂k , byplugging in (7.9) the expressions for the background given by our solutions.Using the expressions for the leading order, first and second subleading orders for themetric and the 5-form given in 5 and in the Appendix one can calculate ∂M∂k = Z S × Y p,q (cid:18) ∂∂k Q gravξ − ξ · θ k (cid:19) = 0 ∂M∂C = Z S × Y p,q (cid:18) ∂∂C Q gravξ − ξ · θ (cid:19) = 2 πL G ∂M∂C = Z S × Y p,q (cid:18) ∂∂C Q gravξ − ξ · θ (cid:19) = − πL G (7.16)where G is the 5-dimensional Planck constant G = G /V ol ( Y p,q ) and θ ia ··· a = 116 πG (cid:2)(cid:0) ∇ b ∂ i g ab − ∇ a ∂ i g bb (cid:1) ǫ aa ··· a − F a ··· a ∂ i A a ··· a (cid:3) (7.17)with ∂ i = ∂∂k , ∂∂C i . (7.18)Putting everything together we conclude that: M = πL G (2 C − C ) = − πL Q G , (7.19)25here we have set the integration constant to zero. Some comments are in order here.First note that the 8-form to be integrated involves directions orthogonal to t and Ω. Therelevant contribution from the 5-form is of the type F (5) t ˆ y ˜1˜2˜3 ∂ i A (4) φ ˆ1ˆ2ˆ3 , which turns out to beof order Ω : ∂ i A (4) φ ˆ1ˆ2ˆ3 goes like Ω , and F (5)Ω φ ˆ1ˆ2ˆ3 = ∂ Ω A (4) φ ˆ1ˆ2ˆ3 ∼ Ω. On the other hand, F (5) t ˆ y ˜1˜2˜3 ,the dual of the latter, goes like Ω − . Therefore the 5-form term gives a finite contributionto ∂ i M . The gravitational contributions to ∂ i M on the other hand contain terms of order1 / Ω , therefore potentially divergent. However the coefficients of these terms turn out to betotal derivatives in the internal coordinates: more precisely, the coefficient is proportionalto ddˆ y q (ˆ y ), therefore it gives vanishing contribution after integrating over ˆ y between the twozeroes of q (ˆ y ), ˆ y and ˆ y . Again this fact can be proven in more generality than just forour BPS solutions [33].Let us now proceed to verify the BPS relation between the mass M and the R-charge R . With our normalisation of the Reeb vector, the BPS relation is given by M L = 32 R (7.20)where R is the charge which sources the KK gauge field A R . The five dimensional equationof motion for its field strength F R are given by τ RR d ⋆ F R = ⋆ J R . (7.21)where J R is the one-form charge current and τ RR comes from the KK reduction. Takingthe integral of the equation of motion, the total charge R can be read from the flux atinfinity of the field strength R = lim ˜ ρ →∞ τ RR Z S (˜ ρ ) F R (7.22)where S ( ˜ ρ ) is the three dimensional sphere in AdS at constant t, ˜ ρ . In Section 5 wederived A R ≈ − Q ρ d t (7.23)at leading order in large ˜ ρ . Following [34] we have τ RR = 316 πG Z g ψ R ψ R vol ( Y p,q ) = 112 πG (7.24)where we have used g ψ R ψ R = as can be seen from (5.17),(5.18). We can now explicitlywrite the value of the total R charge R = − QL πG V ol ( S ) = 23 M L (7.25)which satisfies the expected relation. 26et us mention that we have computed M also using a 5-dimensional definition in-volving the intrinsic 5-dimensional Weyl tensor, due to Ashtekar and collaborators[24] andrederived in [25], H ξ = M = − πG Z S ˜ E tt vol ( S ) (7.26)where ˜ E tt = 12 Ω − ˜ C Ω tt Ω . (7.27)where ˜ C abcd is the Weyl tensor of the unphysical metric ˜ g = Ω g . Beyond leading order AdS and Y p,q coordinates mix, so, in general the metric on the deformed AdS depends onthe choice of the 5-dimensional slice inside the 10-dimensional manifold. The calculation,done using our explicit form of the perturbed metric and allowing a slice dependence on theinternal coordinates, actually reveals that the slice dependence drops out in the Weyl tensorand gives the correct result for the mass, as in the previous 10-dimensional computation.The degree of generality of this result is under investigation [33]. In this paper we have performed an asymptotic, large distance, analysis of 1/2 BPS statesin IIB supergravity
AdS × Y p,q . The corresponding differential equations are the sameas those found in [12], where 1/8 BPS states of IIB supergravity on AdS × S wereanalysed. The difference resides in the boundary conditions, here we require solutionswhich are asymptotic to AdS × Y p,q . They carry non trivial charges under the asymptoticisometries which are dual to the R -charge and one U (1) flavour charges of the quiver gaugetheories. We have shown that the charges are consistent with the holographic principlewhich in this case relates N = 1 quiver gauge theories to gravity on AdS × Y p,q . Of courseour analysis was only asymptotic: we did not address the issue of regularity of the solutionsover the full configuration space. One can analyse the solutions in the opposite regime,near y = 0, but it is difficult to connect this region to the large y region. It would be veryinteresting, although probably quite hard, due to the complexity of the system of non-linear partial differential equations governing them, to prove the existence of non-singularsolutions, which then would be the exact analog of those found in [1] for the maximallysupersymmetric case.In the course of the analysis we had to cope with the problem of defining the mass ofthe states in the asymptotically AdS × Y p,q spacetime. We adopted a ten dimensionalapproach, which uses the definition of charges given by Wald and collaborators. It gives afinite (and correct) result. A different “holographic” approach to this problem, which uses adetailed analysis of the KK reduction of the 10 dimensional theory to AdS can be found in[30, 31, 32]. We had indications, however, that, at least for our backgrounds, an expressiondue to Ashtekar and Das [24, 25], which involves the intrinsic Weyl tensor in the deformed AdS , also gives the correct result. This brings about various questions. For example, aboutthe finiteness of Wald et al. expression, one would like to establish it in general terms,27ithout relying to a particular form of backgrounds. That is, one would like to prove ingeneral, assuming just that the equations of motion hold with the asymptotic behaviour ofthe fields implied by AdS/CFT correspondence, that potentially divergent terms are totalderivatives in the internal compact manifold. Similarly, it would be interesting to see underwhich circumstances the ten dimensional approach finally coincides with the 5-dimensionalone of Ashtekar et al. We hope to come back to these issues in a future publication[33]. A Second order solutions
We give here the complete expression for the second order solutions ρ (2)1 (ˆ y ) = − ( L ( − c ˆ y ) (( − ac +27 k ) ( − c ˆ y − c ˆ y − c ˆ y +11666 c ˆ y − c ˆ y + 16281 c ˆ y − c ˆ y + 1080 c ˆ y + a c ( −
65 + 72 c ˆ y + 20 c ˆ y )++ a c (50+82 c ˆ y +159 c ˆ y − c ˆ y +380 c ˆ y )+ ac ( − − c ˆ y +756 c ˆ y − c ˆ y +6449 c ˆ y + − c ˆ y + 1080 c ˆ y ))+ − − ac )( − ac +27 k )(2+ ac − c ˆ y +3 c ˆ y ) ( − c ˆ y (4 − C ) − c ˆ y (1+4 C )++ 120 c ˆ y (1 + 4 C ) + c ˆ y (133 + 552 C ) + ac (5 − C + 20 c ˆ y (1 + 3 C ) + 2 c (ˆ y + 54ˆ yC )))++ 32( − ac ) (2 + ac − c ˆ y + 3 c ˆ y ) ( −
10 + c ˆ y (2 − C ) + 10 c ˆ y ( − C + 27 C )++ ac ( − c ˆ y (17+41 C )+10 c (ˆ y +3ˆ yC ) +5(3+8 C − C )) − c ˆ y (1+143 C +270 C +30 C )+ c ˆ y (29 + 252 C + 180 C + 90 C )))) / (4320( − ac ) (2 + ac − c ˆ y + 3 c ˆ y ) ) ρ (2)3 (ˆ y ) = ρ (2)1 (ˆ y ) + k (2) (ˆ y ) k (2) (ˆ y ) = ( L ( L − cL ˆ y ) (((1 − c ˆ y )(16( − ac ) ( − c ˆ y )(2 + ac − c ˆ y + 3 c ˆ y ) (11 + 4 ac − c ˆ y + 15 c ˆ y ) − − ac )( − ac + 27 k )(2 + ac − c ˆ y + 3 c ˆ y )( − c ˆ y − c ˆ y +426 c ˆ y − c ˆ y − c ˆ y + a c ( − c ˆ y )+2 ac ( − c ˆ y − c ˆ y ++12 c ˆ y ))+( − ac +27 k ) ( a c (11+ c ˆ y )+6 a c ( − c ˆ y − c ˆ y +7 c ˆ y )+3 ac ( − c ˆ y + − c ˆ y + 200 c ˆ y − c ˆ y + 3 c ˆ y ) + 2( −
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