Closed Strings and Moduli in Ad S 3 /CF T 2
aa r X i v : . [ h e p - t h ] M a y Closed Strings and Moduli in AdS / CFT Olof Ohlsson Sax and Bogdan Stefa´nski, jr. Nordita, Stockholm University and KTH Royal Institute of Technology,Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden Centre for Mathematical Science, City, University of London,Northampton Square, EC1V 0HB, London, UK [email protected],[email protected]
Abstract
String theory on AdS × S × T has 20 moduli. We investigate how the perturbativeclosed string spectrum changes as we move around this moduli space in both the RRand NSNS flux backgrounds. We find that, at weak string coupling, only four of themoduli affect the energies. In the RR background the only effect of these moduliis to change the radius of curvature of the background. On the other hand, in theNSNS background, the moduli introduce worldsheet interactions which enable theuse of integrability methods to solve the spectral problem. Our results show thatthe worldsheet theory is integrable across the 20 dimensional moduli space. ontents A . . . . . . . . . . . . . . . . . . . . . 13 A.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28A.2 IIB supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29A.3 T duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
B General AdS × S × T backgrounds 30 B.1 RR backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30B.2 NSNS backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
C The TrT transformed Green-Schwarz string 32
The AdS / CFT correspondence [1] is an intriguing example of holography. One simplereason for this is the intermediate amount of supersymmetry, which allows rich andinteresting physical phenomena, while at the same time retaining sufficient control forprecise computations. For example, partly because of the low amount of supersymmetry,2he dual pair has a large number of moduli. Perhaps a more significant reason for interestin this dual pair is the keystone role that AdS / CFT plays in diverse areas, such as black-hole entropy [2], the ADHM construction [3] of instantons and their moduli space [4, 5],or holography of two-dimensional maximally-supersymmetric SQCD. For a review of theAdS / CFT correspondence see [6].A compelling feature of AdS / CFT is that the D-brane construction used to con-jecture this duality does not directly provide a CFT living on the brane worldvolume.Rather, the low-energy dynamics of D1/D5 open strings is given by two-dimensionalsuper-Yang-Mills theory (SYM) coupled to adjoint and fundamental matter. In the IRthis theory flows to a direct sum of Coulomb and Higgs branch CFTs [7], with the lat-ter conjectured to be the dual to strings on the near-horizon AdS geometry [1]. Thisbehaviour is in sharp contrast to the more supersymmetric cases where the D3- andM2-brane worldvolume gauge theories are conformal.The lack of a perturbative gauge theory description has made direct tests of theAdS / CFT correspondence more challenging. Based on topological arguments [8], ithas been suggested that the CFT should be closely related to a deformation of a Sym N orbifold [5]. The location of the Sym N orbifold point in the near-horizon moduli spacehas been discussed in [9], but a precise understanding of the relationship between thisorbifold and the full string theory remains to be understood more fully, as discussed forexample on pp. 9-10 of [10]. Nevertheless, many quantities protected by supersymmetryhave been matched between strings on AdS and the Sym N orbifold. However, these testscannot establish the exact nature of the relationship between string theory on AdS andthe Sym N orbifold, precisely because the matching involves only protected quantititesand so does not depend on how, for example, the moduli enter the dictionary. Oneoutstanding problem is to identify precisely the states dual to perturbative closed stringsin the Sym N orbifold.To make progress on this question, we investigate perturbative closed string stateson AdS × S × T with zero winding and zero momentum on T , which we denote by H (0 , . Working in the Green-Schwarz formulation, we determine what effect varyingthe 20 moduli has on the spectrum of H (0 , . As is well known [5], there are many typeIIB string theory backgrounds on the space-time AdS × S × T . These differ from oneanother by the charges that the space-time carries. In this paper we will consider twosuch distinct AdS backgrounds corresponding to the near-horizon limit of D1/D5 andF1/NS5 brane configurations, returning to backgrounds with more general charges inthe future. We find that for both types of AdS backgrounds only 4 moduli affect the H (0 , spectrum. In the D1/D5 background, these 4 are the dilaton φ and the 3 moduliassociated with the self-dual NSNS potential B + . In the F1/NS5 brane background theseare instead the 3 moduli associated to the self-dual part of the RR two-form potential C +2 on T and the modulus associated with a linear combination of C and C . In this paper, we will consider a string theory modulus to be a scalar field that can acquire a vacuumexpectation value without changing the background charges. This should be contrasted with a U-dualitytransformation which will generically change the charges of the background. For example, changing oneof the radii of T is a modulus of the D1/D5-brane or the F1/NS5-brane near-horizon theories. On theother hand, performing an S-duality maps one between the D1/D5 and F1/NS5 backgrounds and soshould not be thought of as a marginal deformation.
3n the AdS theory supported by RR flux, backgrounds with different values of B + are related by T-duality. As a result, the energies of H (0 , states are unchanged apartfrom a modification of the radius of curvature of the theory R = α ′ g s N q ( B + ) , (1.1)where N is the number of D5-branes of the background. Consequently, energies of H (0 , states at any value of these moduli can be determined using the Bethe Equations (BEs)found in [11] by simply modifying R in this way.Turning on the four RR moduli in the AdS theory supported by NSNS flux has amore significant effect on the worldsheet action. As we show, turning on these moduliinduces non-zero RR field-strength couplings and the world-sheet action takes the sameform (up to T-dualities) as the mixed-flux theories considered in [12–14]. The energiesof H (0 , states then follow from the BEs found in [11], using the exact S matricesdetermined in [13, 14], upon a suitable identification of parameters. We show that inthe AdS theory supported by NSNS flux the strength of the integrable interactions,conventionally denoted by h is given by h = − g s c k π + . . . , (1.2)where k is the WZW level, c is the value of the RR moduli, and subleading correctionsin R are denoted by ellipses. Setting the RR moduli to zero turns off the integrableinteractions, which reduces the model to a GS version of the WZW CFT analysed byMaldacena and Ooguri [15]. What is more, by analogy with higher-dimensional holog-raphy we see that (at large R ) c k plays the role of the ’t Hooft coupling constant √ λ ,and one is led to considering suitable double-scaling limits of c and k to capture leading(planar) dynamics.This paper is organised as follows. In section 2 we review the moduli space of stringson AdS × S × T . In sections 3 and 4 we determine the effect of moduli on H (0 , inthe D1/D5 and F1/NS5 backgrounds, respectively. In section 5 we present the completebackreacted geometries with non-zero moduli. We conclude in section 6. The moretechnical aspects of our results are relegated to the appendices. In this section we briefly review the moduli space of Type IIB strings on AdS × S × T ,summarising some of the results of [9]. Type IIB compactified on T , in directions x , . . . , x , has 25 scalars which parametrise the coset SO(5 , × SO(5) . By wrapping D5-branes (respectively, NS5-branes) on T , and D1-branes (fundamental strings) transverseto T but parallel to the other 5-brane directions we obtain the BPS D1/D5 (F1/NS5)background. In the near-horizon limit of both geometries, five of the above scalars get In the integrable literature this is conventionally denoted by h ( λ ) where λ = R /α ′ is the ’tHooft coupling constant. In this paper, we will write h ( R ) rather than h ( λ ), since R is a more naturalparameter from the string theory point of view. × S × T , whichlocally takes the form of a coset K = SO(5 , × SO(4) . (2.1)The U-duality group SO(5 , Z ) [17] acts both on K and on the background charges. Asubgroup of the U-duality group, denoted by H q in [9], leaves the background chargesinvariant, yet acts non-trivially on K through global identifications. As a result [9],globally the moduli space is H q \K . The near-horizon limit of the D1/D5 geometry is AdS × S × T supported by RRthree-form flux, and the 20 moduli are: (i) excluding the overall volume v , (ii) , (iii) B + on T , (iv) the string coupling constant g s , and (v) . In this background the five fixed scalars are (a) , (b) the volume of T , v , and (c) a second linear combination of the RR scalar and four-form potentials.The values that these five scalars take varies with the moduli, and can be determined byminimising the BPS mass-charge formula for the background, which in the case of theD1/D5 system reduces to solving equations (10)-(12) of [9]. For example, at the point G ij = √ vδ ij , C = 0 , B + = 0 , C = 0 , (2.2)where i, j = 6 , , ,
9, we have v = N N , B − = 0 , C = 0 , (2.3)where N and N denotes the number of D1 and D5 branes, respectively. More precise definitions of the moduli will be given in section 3. We will find it convenient to treat the RR scalar C as the modulus and the four form C on T orus as fixed. A more accurate description would be to say that one linear combination of these modesremains massless, and thus is a proper modulus, while the other combination obtains a mass and hencegets a fixed expectation value. .2 Moduli and fixed scalars in the F1/NS5 geometry Similarly, the near-horizon limit of the F1/NS5 geometry is AdS × S × T supportedby NSNS three-form flux, and the 20 moduli are: (i) excluding the overall volume v , (ii) , (iii) , (iv) the string coupling constant g s , and (v) .Here the five fixed scalars are (a) , (b) the volume of T , v , and (c) a second linear combination of the RR scalar and four-form potentials.The values of the fixed scalars are determined by solving equations (15)-(17) of [9]. Forexample, at the point G ij = √ vδ ij , B = 0 , C + = 0 , C = 0 (2.4)we have v = N F1 k g s , C − = 0 , C = 0 , (2.5)where k and N F1 are the number of NS5-branes and fundamental strings, respectively. As reviewed in section 2.1, the near-horizon D1/D5 geometry has 20 moduli. In thissection, we establish what effect these moduli have on energies of H (0 , states. We findthat the spectrum is independent of 16 of them, and determine the influence of theremaining 4 at weak string coupling. It turns out that varying these 4 moduli has aremarkably simple effect on H (0 , . The energy spectrum is determined in terms of theradius of curvature R , which depends on the 4 moduli, as well as on N .Below we investigate how moduli enter the GS action. We find that for 16 modulithe action remains invariant up to simple field redefinitions. Of the remaining 4, one is g s whose only effect is to change R , via equation (1.1). As we show below, the other 3moduli have a non-trivial effect on the action. Luckily, it turns out that this new actionis related to the original one through T-duality. As a result, the H (0 , spectrum at anyvalue of these 3 moduli is determined from the one found in [11, 18] together with asuitable identification of R , which will now depend on these moduli.6 .1 The inconsequential moduli The energies of states in H (0 , do not depend on the 16 moduli (i) , (ii) and (v) ofsection 2.1. It is straightforward to see that the geometric moduli of T have no effecton H (0 , . This is because given any flat metric on T we can always redefine the x i toreduce to the metric (2.2) used in [19]. While such a redefinition changes the periodicityconditions of the x i , this has no effect on the zero-winding zero-momentum states of H (0 , . Next consider turning on a non-zero constant C on T . The gauge-invariant RRfield strengths are defined as F p +1 = dC p − C p − ∧ H, H = dB, (3.1)with H ≡ C has no effect onthe field strengths, which in turn leaves the equations of motion and Bianchi identitiesunchanged, since these depend only on the field strengths rather than potentials (seeequation (A.11)). We may therefore conclude that the AdS × S × T geometry withRR flux can be deformed by turning on a constant C . What is more, the backgroundcharges do not change as we vary C . We can see this explicitly by integrating theequations of motion/Bianchi identities for the fluxes [20]. For example, D5-brane andD3-brane charges are given by Q D5 = 12 κ Z S F = µ N ,Q D3 = 12 κ Z S × T ij F + C ∧ H = 0 , (3.2)neither of which depends on C . Similarly, one can check that the D1-, NS5- and F1-charges remain respectively N , 0 and 0 for all values of C . This shows that turningon a constant C in this geometry corresponds to adjusting the values of the 6 modulilisted in (ii) of section 2.1. The GS action depends only on field strengths and not onpotentials, and so does not change as we vary C . We conclude that the 6 C have noeffect on the energies of states in H (0 , .By an almost identical analysis to the above, one finds that the modulus C also hasno effect on the H (0 , spectrum. Turning on constant values for C and C leaves thefield strengths unchanged and so equations of motions and Bianchi identities continueto be satisfied. Adding constant C and C potentials does change the F1 charge Q F1 = 12 κ Z S × T e − ∗ H + C F + C ∧ F − B ∧ C ∧ F . (3.3) Note that the spectrum is also independent of v , since we can rescale the T coordinates x i → v − / x i . The resulting change in x i periodicity conditions is also inconsequential for H (0 , strings. This is consistent with the fact that a constant RR potential does not induce any additional branecharges in the D1 or D5 brane worldvolume theories. In the case of the D3-brane charge this is because both F and H are identically zero. C is not an independent modulus. Rather it is determined interms of C , B + and C by equation (11) of [9]. This ensures that the F1 charge is zero Q F1 = 12 κ Z S × T e − ∗ H = 0 . (3.4)As above, since the Green-Schwarz action depends only on field strengths rather thanpotentials we conclude that C has no effect on the H (0 , spectrum. In this subsection we show that the energies of states in H (0 , depend on g s and B + listed as (iii) and (iv) in section 2.1. The effect of varying g s is well understood. Theradius of AdS and g s are related by equation (1.1). For g s ≪
1, where perturbativestring theory is valid, varying g s changes the energies of worldsheet excitations throughthe function h ( R ) as described in equation (6.3).At first sight, one might conclude that a constant B + should be just as inconsequentialas a constant C . It is certainly true that adding a constant B + does not change thefield strengths and so gives a consistent family of AdS backgrounds. However, thisdeformation does not correspond to turning on a modulus in the near-horizon D1/D5geometry because the background has a non-zero D3-brane charge Q D3 = 12 κ Z S × T ij F − B + ∧ F = 0 . (3.5)To find a background with Q D3 = 0 we need to further add a non-constant C dC = B + ∧ F . (3.6)The explicit expression for the geometry and fluxes is given in Appendix B, with theAdS radius of curvature R given by R = α ′ e φ N q ( B + ) . (3.7)This geometry corresponds to the near-horizon limit of the fully back-reacted D1/D5system with non-zero B + moduli [22, 23], which we discuss in section 5.1.To recapitulate, turning on B + moduli in the AdS geometry with RR three-formflux induces a non-trivial F flux. This clearly modifies the GS action and so shouldhave a non-trivial effect on the energies of H (0 , states. Fortunately, it is possible toestablish how the spectrum varies with the B + moduli without having to repeat the While a constant C (respectively C ) induces F1 charge on the D1 (D5) worldvolume, a suitablelinear combination of C and C has a trivial net effect. A constant B field induces D3-brane charge in the D5-brane worldvolume theory through its WZcouplings [21]. As a result, we may anticipate that the B + modulus in this background will be moresubtle than C . The resulting F = 0 is self-dual and satisfies its equation of motion. B + modulus is equivalent to performing T-dualities and field redefinitions of theT coordinates together with a compensating shift in the B field. As we show below,understanding the effect of these manipulations on the spectrum is straightforward. Let us briefly review the procedure of [22, 23]. In the undeformed D1/D5 background perform a T-duality along one of the T directions, say x , followed by a redefinition ofthe ( x , x ) variables x → x cos ϕ + x sin ϕ, x → x cos ϕ − x sin ϕ, (3.8)and then a T-duality back along x . We will call this a TrT transformation, since it issimilar to, though distinct from, the TsT transformations, much used in AdS [24, 25].Starting with a square T of area A with no B field, a TrT transformation maps to asquare T of area ˜ A = A (cos ϕ + A sin ϕ ) − / and a B-field ˜ B = (1 − A )(cos ϕ + A sin ϕ ) − / . We can also perform a TrT transformation, parametrised by ψ , in the( x , x ) directions. respectively. The resulting TrT backgrounds carry D1-, D3- andD5-brane charges. However, we can further add a constant B-field˜ B → ˜ B + b , ˜ B → ˜ B + b , (3.9)which, if chosen judiciously [22, 23], (see equation (5.15)) precisely cancels the D3-branecharges of the TrT background. Restricting to ϕ = ψ, (3.10)sets vol( T ) = vol( T ). In summary, turning on the B +67 modulus is equivalent to aTrT deformation with ϕ = ψ and an additional constant B field.Let us now consider the effect of these manipulations on the worldsheet action. Aconstant shift of the B field (3.9), has no effect on the GS action since this depends onlyon field strengths. While such a B field does modify the periodicity conditions of theT coordinates x i , this has no effect on the zero-momentum, zero-winding strings thatwe are interested in. As a result, the H (0 , spectrum of the theory deformed by a B + modulus is the same as the H (0 , spectrum of the TrT theory with ψ = ϕ . In turn, the H (0 , spectrum of the TrT theory can be obtained from the original undeformed theoryby analysing the effects of T-dualities and field redefinitions, as we now describe.The T coordinates x i enter the GS action only through their derivatives, since theycorrespond to U (1) isometries S = − π Z d σ (cid:18) γ αβ ∂ α x i ∂ β x j G ij − ǫ αβ ∂ α x i ∂ β x j B ij + 2 ∂ α x i (cid:16) γ αβ U β,i − ǫ αβ V β,i (cid:17) + L rest (cid:19) . (3.11) See appendix 5.1 for a presentation of the D1/D5 system in our conventions. Explicit expressions for the metric and other fields in are presented in section 5.2.
G, B, V and U can be found in [19] to quadratic order in fermions. The Noether currents for the x i shift symmetries are J αi ( x ) = − (cid:16) γ αβ ∂ β x j G ij − ǫ αβ ∂ β x j B ij + γ αβ U β,i − ǫ αβ V β,i (cid:17) . (3.12)The TrT-transformed action takes the same general form as (3.11), for the dual coordi-nates ˜ x i with couplings ˜ G , ˜ B , ˜ U and ˜ V . The corresponding Noether currents are˜ J αi (˜ x ) = − (cid:16) γ αβ ∂ β ˜ x j ˜ G ij − ǫ αβ ∂ β ˜ x j ˜ B ij + γ αβ ˜ U β,i − ǫ αβ ˜ V β,i (cid:17) , (3.13)and the coordinates x , and ˜ x , are related via ∂ α ˜ x = cos ϕ ∂ α x − sin ϕ (cid:16) ǫ αβ γ βδ ∂ δ x i G i − ∂ α x i B i − ǫ αβ γ βδ U δ, − V α, (cid:17) ,∂ α ˜ x = cos ϕ ∂ α x + sin ϕ (cid:16) ǫ αβ γ βδ ∂ δ x i G i − ∂ α x i B i − ǫ αβ γ βδ U δ, − V α, (cid:17) . (3.14)As a result, the currents J αi and ˜ J αi are B¨acklund transforms of each other˜ J α = cos ϕ J α − sin ϕ ǫ αβ ∂ β x , ˜ J α = cos ϕ J α + sin ϕ ǫ αβ ∂ β x , (3.15)and we have ∂ σ ˜ x = cos ϕ ∂ σ x − sin ϕ J τ , ∂ σ ˜ x = cos ϕ ∂ σ x + sin ϕ J τ . (3.16)Integrating the above with respect to σ and reasoning in a similar way to [25], wefind that instead of analysing states with conventional periodicity on a TrT-transformedbackground ˜ x (2 π ) − ˜ x (0) = ˜ w , ˜ x (2 π ) − ˜ x (0) = ˜ w , (3.17)we can consider states in the untransformed background with twisted periodicity condi-tions for x , x (2 π ) − x (0) = sec ϕ ˜ w + P tan ϕ, x (2 π ) − x (0) = sec ϕ ˜ w − P tan ϕ. (3.18)Above, P , are charges (momenta) of the currents J α , . What is more, since J τi is themomentum variable conjugate to x i , we can in fact relate the TrT-transfomed variablesto the original ones through a canonical transformation˜ p = cos ϕ p − sin ϕ x ′ , ˜ x ′ = cos ϕ x ′ − sin ϕ p , ˜ p = cos ϕ p + sin ϕ x ′ , ˜ x ′ = cos ϕ x ′ + sin ϕ p . (3.19)We therefore conclude that the spectrum of closed strings in a TrT-transformed back-ground will be the same as the spectrum of strings with twisted periodicity condi-tions (3.18) in the original background. This is true for all perturbative closed stringstates, since the TrT transformation is an exact T-duality symmetry of the theory. As a result, the fermions are automatically uncharged under the U (1) isometries of T , unlike whathappens in TsT backgrounds [26]. Explicit expressions for ˜ G , ˜ B , ˜ U and ˜ V can be found in appendix C.
10n the H (0 , sector of most interest in this paper, states have zero winding andmomentum, and it is easy to see that equation (3.18) maps periodic states to periodicstates. As a result, the energies of H (0 , states before and after TrT transformations arethe same. At weak coupling, they depend only on the respective radii of curvature ofthe two theories, which are equal to one another, and taking into account the shift ofthe dilaton under T-duality can be written as R = α ′ e φ N = α ′ e ˜ φ (cid:16) N cos ϕ + N sin ϕ (cid:17) . (3.20)Note that this expression is written in terms of the original D-brane charges N and N .This expression gets further corrected by the constant shift of the B field, as discussedin section 5.2. From the arguments given above equation (3.11), we then find that theenergies of H (0 , states depend on the value of the B + moduli through the dependenceof R in equation (3.7).In [11,18] the energies of H (0 , strings were found as solutions of (essentially algebraic)BEs. The explicit calculations were carried out at the point in moduli space given inequation (2.2) and at small g s . It was shown that the 2-to-2 worldsheet S matrix isfixed by symmetries alone [19, 27–29]. and satisfies the Yang-Baxter equation. Thisdetermines the complete worldsheet scattering and hence the spectrum through the BetheAnsatz.In principle, this concludes our analysis of the dependence of the H (0 , spectrum on g s and B + . In the remainder of this section, we show more explicitly that deriving the 2-to-2S matrix following [19], is fully compatible with the TrT deformations we have discussedabove. We pay particular attention to gauge-fixing, expressions for supercharges and theoff-shell algebra A under TrT. The reader not interested in these technical details maywish to proceed directly to the next section.T-duality along T directions can be used to map the background considered here toother Type IIB backgrounds: the D3/D3’ background or the D1/D5 background with N and N swapped, or related Type IIA backgrounds. The energies of H (0 , stateswill remain unchanged under such T-dualities and since the states carry no winding ormomentum T-duality will map them to H (0 , states in the dual background. The string two-body S matrix found in [19] was determined in a particular gauge. It istherefore worth checking that the gauge-fixing is compatible with TrT transformations.The gauge fixing is a two-step process: fixing kappa gauge, and fixing uniform light-conegauge.In [19] a kappa-gauge that is particularly well adapted to the underlying integrabilitywas used. This kappa-gauge is a simple projection on the (spectator) fermions, and asresult it commutes with T-duality and redefinitions of the T bosons. We can therefore More precisely symmetries fix the matrix part of the S matrix, leaving undetermined overall nor-malisations known as dressing factors. The dressing factors are then determined by solving suitablecrossing equations [11, 30]. x i and˜ x i equations of motion are given by current conservation equations ∂ α J α = 0 , ∂ α ˜ J α = 0 , (3.21)and are equivalent to one another when equation (3.14) is used. The equations of mo-tion for the other fields, including the Virasoro constraints for the worldsheet metric γ αβ , remain the same in the original and TrT-transformed background upon using therelations between G, B, U, V and ˜ G, ˜ B, ˜ U, ˜ V , as well as equation (3.14). To see this, letus introduce a collective field ω ( τ, σ ) = n x ± , y i , z i , χ, η, ∂ α x ± , ∂ α y i , ∂ α z i , ∂ α χ, ∂ α η o . (3.22)Then one can show that γ αβ ∂ α x i ∂ β x j ∂G ij ∂ω − ǫ αβ ∂ α x i ∂ β x j ∂B ij ∂ω + 2 ∂ α x i γ αβ ∂U β,i ∂ω − ǫ αβ ∂V β,i ∂ω ! + L rest ∂ω = γ αβ ∂ α ˜ x i ∂ β ˜ x j ∂ ˜ G ij ∂ω − ǫ αβ ∂ α ˜ x i ∂ β ˜ x j ∂ ˜ B ij ∂ω + 2 ∂ α ˜ x i γ αβ ∂ ˜ U β,i ∂ω − ǫ αβ ∂ ˜ V β,i ∂ω ! + ˜ L rest ∂ω . (3.23)Hence the equations of motion for the non-T sigma-model fields are the same beforeand after a TrT transformation. Similarly, one can show that δSδγ αβ = δ ˜ Sδγ αβ , (3.24)confirming that the Virasoro constraints of the two theories are also the same.In uniform light-cone gauge [19, 31] we set x + ≡ ∂ α ( φ − t ) / τ and its conjugatemomentum, p − = 1. The Virasoro constraints then determine the non-dynamical field x − ≡ ∂ α ( φ − t ) / γ αβ ,while the x ± equations of motion are used to fix γ αβ . As we have shown above the x ± and γ αβ equations of motion are invariant under TrT transformations. As a result,gauge-fixing the TrT-transformed action gives the same expressions for ˙ x − , ′ x − and γ αβ asthose found in the original background [19]; the dependence on ∂ α x i can be re-expressedin terms of ∂ α ˜ x i by using equation (3.14), as well as the relations between G, B, U, V and ˜ G, ˜ B, ˜ U, ˜ V . We conclude that gauge-fixing the GS action is compatible with TrTtransformations. In [19], supercurrents Q α were constructed, in terms of transverse fields and their deriva-tives, as well as the ubiquitous non-local prefactor e ± ix − . The conservation of thesesupercurrents was checked using the equations of motion. In these expressions for thesupercurrents the torus bosons x i enter only through the first derivatives ∂ α x i .12herefore, in order to find the supercharges in the TrT theory, one can use theinverse of equation (3.14) to re-express Q α in terms of the transverse fields of the TrT-transformed theory, as well as the pre-factor e ± ix − . As we reviewed in the previoussub-section, x − is determined via the Virasoro constraints, which are the same in theTrT-related theories upon using the relations between G, B, U, V and ˜ G, ˜ B, ˜ U, ˜ V , as wellas equation (3.14). To summarise, we can express the supercurrents Q α , exclusively interms of the fields that enter the TrT transformed action ˜ S . We will denote by ˜ Q α thisputative supercurrent.It remains to be checked whether ˜ Q α is conserved when equations of motion derivedfrom ˜ S are used. This however, has to be, because: (i) the equations of motion for allfields other than ˜ x i (including γ αβ ) are the same in the two theories; (ii) the ˜ x i equationsof motion together with equation (3.14) are equivalent to the x i equations of motion.Therefore, ˜ Q α is conserved upon using equations of motion derived from ˜ S . A Finally, we turn to the algebra A of supercharges which commute with the Hamiltonian.In order to find the commutation relations of A in the undeformed theory, it was neces-sary to redefine the fermions in order to obtain a canonical kinetic term for the fermions;see Appendix I of [19] for details. In principal, an analogous computation should beperformed in the TrT transformed background. Explicitly one needs to find δ ˜ Sδ ¯ ηδ ˙ η , δ ˜ Sδ ¯ χδ ˙ χ , δ ˜ Sδ ¯ χδ ˙ η , δ ˜ Sδ ¯ ηδ ˙ χ . (3.25)It is easy to see that the first two expressions involve only ˜ L rest . However, to quadraticorder in fermions ˜ L rest = L rest , hence equations (I.1) and (I.2) of [19] remain the sameafter TrT transformations. On the other hand the mixed η - χ terms at first appear tochange after a TrT transformation. We note that these terms do not involve ˜ L rest , andto the order that we are working to ∂ ˜ G∂ ¯ χ∂ ˙ η = ∂ ˜ B∂ ¯ χ∂ ˙ η = 0 . (3.26)As a result, the only non-zero contributions to η - χ couplings come from ˜ U and ˜ V terms,and to the order we are working these give − π δ ˜ Sδ ¯ χδ ˙ η = ∂ α x i γ αβ ∂ ˜ U β,i ∂ ¯ χ∂ ˙ η − ∂ α x i ǫ αβ ∂ ˜ V β,i ∂ ¯ χ∂ ˙ η , − π δ ˜ Sδ ¯ ηδ ˙ χ = ∂ α x i γ αβ ∂ ˜ U ( f ) β,i ∂ ¯ η∂ ˙ χ − ∂ α x i ǫ αβ ∂ ˜ V ( f ) β,i ∂ ¯ η∂ ˙ χ . (3.27)Given the trivial contributions of ˜ G , ˜ B to the mixed terms, and the fact that γ αβ doesnot depend on fermions, at quadratic order in fermions the above expressions can be13e-written as − π δ ˜ Sδ ¯ χδ ˙ η = ∂∂ ˙ η " γ αβ ∂ α ˜ x i ∂ β ˜ x j ∂ ˜ G ij ∂ ¯ χ − ǫ αβ ∂ α ˜ x i ∂ β ˜ x j ∂ ˜ B ij ∂ ¯ χ + 2 ∂ α ˜ x i γ αβ ∂ ˜ U β,i ∂ ¯ χ − ǫ αβ ∂ ˜ V β,i ∂ ¯ χ ! + ˜ L rest ∂ ¯ χ , − π δ ˜ Sδ ¯ ηδ ˙ χ = ∂∂ ˙ χ " γ αβ ∂ α ˜ x i ∂ β ˜ x j ∂ ˜ G ij ∂ ¯ η − ǫ αβ ∂ α ˜ x i ∂ β ˜ x j ∂ ˜ B ij ∂ ¯ η + 2 ∂ α ˜ x i γ αβ ∂ ˜ U β,i ∂ ¯ η − ǫ αβ ∂ ˜ V β,i ∂ ¯ η ! + ˜ L rest ∂ ¯ η . (3.28)We now observe that the expressions inside the square-brackets above are precisely thesame as the right hand side of equation (3.23) with ω = ¯ χ, ¯ η . Therefore, using equa-tion (3.23) we conclude that δ ˜ Sδ ¯ χδ ˙ η = δ Sδ ¯ χδ ˙ η , δ ˜ Sδ ¯ ηδ ˙ χ = δ Sδ ¯ ηδ ˙ χ . (3.29)In other words, equation (I.3) of [19] remains unchanged As a result, the redefinitionof fermions in the TrT-transformed background in order to obtain a canonical kineticterm, is the same as the one used in Appendix I of [19]. From this we finally concludethat the commutation relations for A are also the same as [19]. The near-horizon F1/NS5 geometry has 20 moduli, which we summarised in section 2.2.We would like to understand their effect on energies of zero-winding zero-momentumclosed string states, which we will continue to denote by H (0 , . Here too, we find thatthe spectrum is independent of 16 of them, and determine the influence of the remaining4 at small g s . As in the previous section, our analysis will primarily rely on the effectthat the moduli have on the worldsheet Green-Schwarz action. The energies of states in H (0 , do not depend on the 16 moduli (i) , (ii) and (iv) ofsection 2.2. As in the D1/D5 background, the geometric moduli of T have no effecton H (0 , , because they can be absorbed into suitable redefinitions of the x i . The stringaction and periodicity conditions of H (0 , are also independent of vol(T ) as before (seefootnote 5). The string coupling constant g s and vol(T ) are related to one anotherthrough equation (17) of [9]. Since g s enters the GS action only through this relation tovol( T ), we conclude that at small string coupling g s has no effect on H (0 , energies. Recall that one uses equation (3.14) to swap between x i and ˜ x i . B = 0 gives a consistent background since H is unchangedand the gauge-invariant RR field strengths (3.1) remain equal to zero. One can easilycheck that the F1 and NS5 charges of the background are unchanged Q NS5 = 12 κ Z S H = µ k,Q F1 = 12 κ Z S × T e − ∗ H + C F + C ∧ dC + H ∧ C ∧ C = µ N F1 . (4.1)and that the D5-, D3- and D1-brane charges are zero Q D5 = 12 κ Z S F + C H = 0 ,Q D3 = 12 κ Z S × T ij F + C ∧ H = 0 ,Q D1 = 12 κ Z S × T F + C ∧ H = 0 , (4.2)because the RR potentials and RR gauge-invariant field strengths are all zero. In otherwords, turning on the B moduli is accomplished in the geometry by setting B to a non-zero constant on T . The GS action depends only on gauge-invariant field strengths ( H and F p ) and so does not change as we vary B . We conclude that the 6 B moduli haveno effect on H (0 , energies. In this subsection we show that the energies of states in H (0 , depend on C and C +2 listed as (iii) and (v) in section 2.2. In fact, turning on a particular C +2 modulus, isequivalent to turning on C and C , as can be seen by performing two T-dualities on T .As a result, we will first focus on turning on just the C modulus.Let us then consider a background with constant RR potentials C = c , C = − c e ∧ e ∧ e ∧ e . (4.3)Since H is non-vanishing, this gives rise to non-zero RR three- and seven-form fieldstrengths F = dC − C H = − c H = − c k (cid:16) Ω AdS + Ω S (cid:17) ,F = dC − C ∧ H = c k (cid:16) Ω AdS + Ω S (cid:17) ∧ e ∧ e ∧ e ∧ e . (4.4) Note that we let the RR potential C take arbitrary real values. Using the SL(2 , Z ) symmetry oftype IIB string theory we can shift C by an integer: C → C − n . This transformation also acts on thepotentials C and C as a shift C → C − nB and C → C − n B ∧ B . Together, these transformationsleave the gauge invariant field strengths F and F unchanged, but shift the D-brane charges. Wetherefore prefer to keep the charges fixed and not put any constraints on the value of C .
15t is straightforward to check that the D5-, D3- and D1-brane charges are all zero Q D5 = 12 κ Z S F + C H = 12 κ Z S − C H + C H = 0 ,Q D3 = 12 κ Z S × T ij F + C ∧ H = 0 ,Q D1 = 12 κ Z S × T F + C ∧ H = 12 κ Z S × T − C ∧ H + C ∧ H = 0 . (4.5)The NS5-brane charge remains unchanged, since H stays the same, as does the F1 charge Q F1 = 12 κ Z S × T e − ∗ H + C F + C ∧ dC + H ∧ C ∧ C = 12 κ Z S × T e − ∗ H + C F = k ( g − s + c ) vol(T ) = µ N F1 . (4.6)The last equality follows from equation (17) of [9], which determines vol(T ) in terms ofthe moduli. In summary, turning on a constant value of the C modulus is implementedin the geometry not just through constant RR zero- and four-form potentials (4.3), butalso through induced three- and seven-form RR field strengths (4.4), with the AdS radius of curvature R given by R = α ′ k q g s c . (4.7)The non-zero RR field strengths (4.4) have an important consequence on the GSaction in this background: the world-sheet action takes the same form as the actionused to analyse mixed flux backgrounds [13, 14]. This is because, from the point of viewof the GS action, a non-zero F generated by a non-trivial C , or by C H are completelyequivalent. As a result, the exact worldsheet S matrix found in [13, 14] applies directlyto the analysis of the H (0 , spectrum of the F1/NS5-brane theory deformed by the C modulus. We simply need to relate the parameters used there to those used here!In [13, 14] the fluxes are e φ F = ˜ q (cid:16) Ω AdS + Ω S (cid:17) , H = q (cid:16) Ω AdS + Ω S (cid:17) , (4.8)together with the condition q + ˜ q = 1. So, replacing˜ q → − g s c k α ′ R , q → k α ′ R , (4.9) This geometry corresponds to the near-horizon limit of a fully back-reacted F1/NS5 system withnon-zero C , which we obtain by U-duality from the solutions [22, 23] in section 5.4. The physical interpretation of the two backgrounds is of course different. The mixed flux back-ground investigated in [13, 14] corresponds to the near-horizon limit of non-threshold bound states ofF1/NS5- and D1/D5-branes, while the background studied in this section is a marginal deformation ofthe F1/NS5-brane near-horizon geometry. radius is given in equation (4.7) above, we obtain the complete S matrixof the C deformation of the pure NSNS AdS geometry.The H (0 , spectrum consists of the BMN vacuum | i BMN on top of which we can actwith magnon-like creation operators denoted schematically as α I p † . . . α I K p K † | i BMN , (4.10)where the indices I , . . . , I K label the excitations above the BMN vacuum [32]. Each ofthe α I p † carries a momentum p i and has an energy E ( p i ) = r(cid:16) m i + kp i π (cid:17) + 4 h ( R ) sin (cid:16) p i (cid:17) , (4.11)with the total energy of a state being the sum of the magnon energies. The strength ofthe worldsheet interactions is governed by the function h ( R ), which in the mixed-fluxbackgrounds took the form h ( R ) = ˜ q π R α ′ + O ( R ) . (4.12)Therefore, for the background obtained by a C marginal deformation of the pure NSNSflux theory, we have h ( R ) = − g s c k π + O ( R ) . (4.13)Notice that something rather remarkable happens: the strength of the worldsheet inter-actions is now proportional to c , and it is this parameter that plays the analogue of the’t Hooft coupling λ that conventionally interpolates between the weakly and stronglycoupled regimes. At small c the interactions are weak, with the dispersion relationbecoming linear in the c going to zero limit. As c increases, the interactions becomemore important, modifying the dispersion relation. Throughout this range the magnonmomenta p i satisfy Bethe Equations derived in [33]. The above conclusions are all validto leading order in the large- R limit, and we expect the function h ( R ) to receive cor-rections when R becomes small. It would be interesting to understand these in order toconnect to the recent investigations of the k = 1 theory [34].So far we have considered the case of a single modulus turned on. In general we canturn on any combination of a constant C and a constant and self-dual C . By a rescalingand rotation of the torus directions we can always align C so that the non-vanishingcomponents point in directions 67 and 89. Hence we are lead to consider a solution with C = c , C = c (cid:16) e ∧ e + e ∧ e (cid:17) , C = − c e ∧ e ∧ e ∧ e . (4.14)This solution will have non-trivial RR field strengths F = − C H, F = − C ∧ H, F = − C ∧ H, (4.15)but does not carry any RR charges. In order to analyse this background we perform aTrT transformation in the directions x and x with a rotation angle of ϕ . This results It is worth noting that marginal deformations of the WZW model on S have been studied in thepast using transformations similar to TrT and TsT transformations [35]. These differ from the RRdeformations considered here because of their non-trivial effect on the S metric.
17n a new background of the same type but where the RR potentials are now given by˜ C = c cos ϕ + √ vc sin ϕ, ˜ C = ( c cos ϕ − √ vc sin ϕ )(˜ e ∧ ˜ e + ˜ e ∧ ˜ e ) , ˜ C = − ( c cos ϕ + √ vc sin ϕ )˜ e ∧ ˜ e ∧ ˜ e ∧ ˜ e , (4.16)where v is volume of the original T . If we choose the angle ϕ so thattan ϕ = c √ vc , (4.17)the ˜ C potential vanishes and we are left with a background of the type discussed earlierin this section, but with the value of the modulus and the AdS and S radii taking thevalues ˜ c = ( c + c ) √ v q c + vc , ˜ R = α ′ k q g s ( c + c ) , (4.18)where g s is the string coupling before the TrT transformation. In the previous sections we have discussed how the closed string spectrum of the near-horizon geometry AdS × S × T is affected when the moduli of the background are turnedon. We will now see how these moduli can be introduced in the full backreacted andasymptotically flat brane geometry. Following [22] and [23] we will start with the D1/D5system and introduce a B field by applying two TrT transformations, as well as adding aconstant B in order to cancel the resulting D3 charges. From the resulting backgroundwe can then obtain other F1/NS5 and D1/D5 backgrounds with non-trivial moduli by Uduality. The corresponding near-horizon geometries are discussed in appendix B. Herewe present the values of the parameters of the near-horizon solutions in terms of thephysical brane charges. Let us start by writing down a type IIB supergravity solution corresponding to thestandard D1/D5 system. We consider a stack of N D1 branes stretched along thedirections ζ and ξ , and a stack of N D5 branes along ζ , ξ , x , x , x and x , wherethe last four directions are compactified on a T . The metric of the full D1/D5 systemis given by ds = ( f f ) − / (cid:16) − dζ + dξ (cid:17) + ( f f ) / (cid:16) dρ + ρ ds (cid:17) + (cid:18) f f (cid:19) / (cid:16) dx + dx + dx + dx (cid:17) , (5.1) In order to not confuse the coordinates for the D1/D5 system with those of AdS , we use thecoordinates ζ , ξ and ρ in the asymptotically flat geometry, and reserve t , z and r in the near-horizonlimit. f = 1 + α ′ ν ρ , f = 1 + α ′ ν ρ . (5.2)The D1/D5 system has a non-trivial dilatonΦ = 12 log f f , (5.3)and is supported by a RR three-form F = − df − ∧ dζ ∧ dξ + 2 α ′ ν Ω S . (5.4)Note that in the asymptotic region, where ρ → ∞ , the metric (5.1) becomes flat, withthe three sphere having unit radius, and the T having unit volume. Furthermore, thedilaton is normalised so that it vanishes asymptotically.The corresponding D1 and D5 charges are give by Q D5 = 12 κ Z S F = ν (2 π ) ( α ′ ) = µ N ,Q D1 = 12 κ Z S × T ∗ F = ν πα ′ = µ N , (5.5)where µ p = (2 π ) − p ( α ′ ) − ( p +1) / is the charge density of the D p brane. From this we findthat the parameters ν and ν are related to the number of branes by ν = N , ν = N . (5.6) Near-horizon geometry.
In the near-horizon limit, the D1/D5 system geometry be-comes AdS × S × T . The AdS and S radii R and the dilaton and T volume are nowgiven by R = α ′ q N N = α ′ e Φ N , e = vol(T ) = N N , (5.7)while the other parameters are turned off. To turn on a B field on T we employ the same strategy as was discussed previously inthe near-horizon geometry: we perform a TrT transformation, and add a constant Bwhich can be adjusted so that there is no D3 charge.The resulting metric takes to form d ˜ s = ( f f ) − / (cid:16) − dζ + dξ (cid:17) + ( f f ) / (cid:16) dρ + ρ ds (cid:17) + ( f f ) / f − ϕ (cid:16) d ˜ x + d ˜ x (cid:17) + ( f f ) / f − ψ (cid:16) d ˜ x + d ˜ x ) (cid:17) , (5.8)19nd the dilaton is given by ˜Φ = 12 log f f f ϕ f ψ , (5.9)where f ϕ = 1 + α ′ ν ϕ ρ , ν ϕ = ν cos ϕ + ν sin ϕ,f ψ = 1 + α ′ ν ψ ρ , ν ψ = ν cos ψ + ν sin ψ. (5.10)Introducing the three forms K = − df − ∧ dζ ∧ dξ + 2 α ′ ν Ω S , ˜ K = − df − ∧ dζ ∧ dξ + 2 α ′ ν Ω S , (5.11)we can write the other non-trivial background fields as˜ F = cos ϕ cos ψ ˜ K − sin ϕ sin ψK , ˜ F = − f − ϕ (cid:16) f cos ϕ sin ψK + f sin ϕ cos ψ ˜ K (cid:17) ∧ d ˜ x ∧ d ˜ x − f − ψ (cid:16) f sin ϕ cos ψK + f cos ϕ sin ψ ˜ K (cid:17) ∧ d ˜ x ∧ d ˜ x . ˜ B = − (cid:16) f − ϕ ( f − f ) cos ϕ sin ϕ − b (cid:17) d ˜ x ∧ d ˜ x − (cid:16) f − ψ ( f − f ) cos ψ sin ψ − b (cid:17) d ˜ x ∧ d ˜ x (5.12)Let us now calculate the various D p -brane charges carried by this solution, starting withthe D3 charges, which are given by Q = 12 κ Z S × T (cid:16) ˜ F − ˜ B ∧ ˜ F (cid:17) , Q = 12 κ Z S × T (cid:16) ˜ F − ˜ B ∧ ˜ F (cid:17) . (5.13)These charge densities are well-defined in the sense that they are given in terms of globallydefined forms, and are independent of the transverse radial coordinate. Performing theintegrals we get µ − Q = ν sin ϕ ( b sin ψ − cos ψ ) − ν cos ϕ ( b cos ψ + sin ψ ) ,µ − Q = ν sin ψ ( b sin ϕ − cos ϕ ) − ν cos ψ ( b cos ϕ + sin ϕ ) . (5.14)The vanishing of the D3-brane charges then leads to b = ν cos ϕ sin ψ + ν sin ϕ cos ψν sin ϕ sin ψ − ν cos ϕ cos ψ , b = ν sin ϕ cos ψ + ν cos ϕ sin ψν sin ϕ sin ψ − ν cos ϕ cos ψ . (5.15)From now on we will impose the above relations.The D1 and D5 charges of the transformed background are given by Q D5 = 12 κ Z S ˜ F = µ ˜ ν = µ ˜ N , (5.16)20nd Q D1 = 12 κ Z S × T (cid:16) ∗ ˜ F + ˜ B ∧ ˜ F − ˜ B ∧ ˜ B ∧ ˜ F (cid:17) = µ ν ν ˜ ν = µ ˜ N , (5.17)where ˜ ν = ν cos ϕ cos ψ − ν sin ϕ sin ψ. (5.18)The above relations can be used to express the parameters ν and ν in terms of thephysical quantities ˜ N and ˜ N . Near-horizon geometry.
In the near-horizon limit, the transformed background isstill given by AdS × S × T , with the AdS and S radii, dilaton and T volume givenby R = α ′ √ ν ν = α ′ q ˜ N ˜ N , e = vol(T ) = ν ν ν ϕ ν ψ = ˜ N ˜ N + ˜ N sin ( ϕ + ψ ) . (5.19)The near-horizon B field can be written as B = b ( e ∧ e + e ∧ e ), with b = − vuut ˜ N ˜ N sin( ϕ + ψ ) . (5.20)We can the write the radius R and the T volume as R = α ′ e ˜Φ ˜ N √ b , vol(T ) = 11 + b ˜ N ˜ N . (5.21)In this form the dependence on the moduli becomes manifest. Let us now apply S duality to the TrT transformed D1/D5 system. We find the metric ds = q f ϕ f ψ f f (cid:16) − dζ + dξ (cid:17) + q f ϕ f ψ (cid:16) dρ + ρ ds (cid:17) + vuut f ψ f ϕ (cid:16) dx + dx (cid:17) + vuut f ϕ f ψ (cid:16) dx + dx (cid:17) . (5.22)The dilaton is given by Φ = 12 log f ϕ f ψ f f . (5.23)The geometry is supported by the NSNS three form H = cos ϕ cos ψ ˜ K − sin ϕ sin ψK . (5.24) Note that the TrT transformation changes the charge quantisation condition so that we now shouldexpress the parameter ν and ν in terms of new integer charges ˜ N and ˜ N . F = − f − ϕ (cid:16) f cos ϕ sin ψK + f sin ϕ cos ψ ˜ K (cid:17) ∧ d ˜ x ∧ d ˜ x − f − ψ (cid:16) f sin ϕ cos ψK + f cos ϕ sin ψ ˜ K (cid:17) ∧ d ˜ x ∧ d ˜ x . (5.25)as well as the RR two-form potential C = (cid:16) f − ϕ ( f − f ) cos ϕ sin ϕ − b (cid:17) d ˜ x ∧ d ˜ x (cid:16) f − ψ ( f − f ) cos ψ sin ψ − b (cid:17) d ˜ x ∧ d ˜ x . (5.26)This potential does not lead to any D1-brane or D5-brane charges. To see that therealso is no D3-brane charges we compute12 κ Z S × T (cid:16) F + C ∧ H (cid:17) = 0 . (5.27)Hence, the only non-vanishing charges are the NS5 charge Q NS5 = 12 κ Z S H = µ ˜ ν. (5.28)and the F1 charge Q F1 = 12 κ Z S × T (cid:16) e − ∗ H + C F + C ∧ C ∧ H (cid:17) = µ ν ν ˜ ν , (5.29)Note that these charges take the same values as the D5 and D1 charges in equations (5.16)and (5.17). Near-horizon geometry.
In the near-horizon limit, the AdS and S radii are givenby R = α ′ √ ν ν = α ′ ˜ N vuut N ˜ N sin ( ϕ + ψ ) , (5.30)where ˜ N and ˜ N now count the number of fundamental strings and NS5 branes. Thedilaton takes the form e = ν ϕ ν ψ ν ν = ˜ N ˜ N N ˜ N sin ( ϕ + ψ ) ! . (5.31)The volume of the torus vol(T ) = 1 is constant in the full backreacted geometry andthus remains the same in the near horizon limit.The RR two-form potential can be written as C = c ( e ∧ e + e ∧ e ) with c = ˜ N ˜ N sin( ϕ + ψ ) r ˜ N ˜ N sin ( ϕ + ψ ) . (5.32)We can the write the radius R and the torus volume directly in terms of physical pa-rameters as R = α ′ ˜ N q e c , vol(T ) = e e c ˜ N ˜ N . (5.33)22 .4 The F1/NS5 system with a RR scalar and four form We finally apply T duality transformations along directions x and x . We then obtainthe metric ds = q f ϕ f ψ f f (cid:16) − dζ + dξ (cid:17) + q f ϕ f ψ (cid:16) dρ + ρ ds (cid:17) + vuut f ψ f ϕ (cid:16) dx + dx + dx + dx (cid:17) . (5.34)The NSNS field strength remains the same as in the previous case, H = cos ϕ cos ψ ˜ K − sin ϕ sin ψK . (5.35)The turned on modulus is encoded in the RR scalar, which takes the form C = c − ν − ν ν ϕ sin 2 ϕf ϕ , (5.36)where c = − ν ν ν ϕ ˜ ν sin( ϕ + ψ ) . (5.37)Also the RR four-form potential is turned on C = (cid:18) − c + ν − ν ν ϕ ν cos ϕ sin ψ − ν sin ϕ cos ψf ψ (cid:19) e ∧ e ∧ e ∧ e . (5.38)Finally, the RR three-form field strength is given by F = f f ϕ cos ϕ cos ψK + f f ψ sin ϕ cos ψ ˜ K . (5.39)While this solution has several non-vanishing RR field strengths, all the D brane chargesare zero. The F1 and NS5 charges are the same as in equations (5.29) and (5.28). Near-horizon geometry.
In the near-horizon limit, the AdS and S radii are givenby R = α ′ √ ν ν = α ′ ˜ N vuut N ˜ N sin ( ϕ + ψ ) , (5.40)and the dilaton and torus volume by e Φ = ν ϕ ν ν , vol(T ) = ν ϕ ν ψ . (5.41)The RR scalar and four form potential are constant in the near-horizon limit, C = c , C = − c e ∧ e ∧ e ∧ e , (5.42)where c is defined in equation (5.37). In terms of this parameter we can write R = α ′ ˜ N q e c , vol(T ) = e e c ˜ N ˜ N . (5.43)23 .5 The D1/D5 system with RR scalar and four form We can finally make an S duality transformation to get back to a D1/D5 backgroundwith a RR scalar and four-form potential. The resulting metric takes the form ds = 1˜ ν q ˜ f f ψ f f ( − dζ + dξ ) + q ˜ f f ψ ˜ ν ( dρ + ρ ds )+ 1˜ ν vuut ˜ ff ψ ( dx + dx + dx + dx ) , (5.44)where ˜ ν was introduced in (5.18) and˜ f = ν cos ψf + ν sin ψf . (5.45)The dilaton and the T volume are now given by e Φ = ˜ f √ f f ˜ ν , vol(T ) = ˜ ff ψ ˜ ν . (5.46)The background is supported by the RR three-form F = ˜ νν ψ f ψ f f ˜ f d ˜ f ∧ dζ ∧ ξ + 2 α ′ ˜ ν Ω S . (5.47)There is furthermore a B field B = ( f − − f − ) r sin(2 ψ )2 α ′ ˜ ν dζ ∧ dξ, (5.48)a RR scalar C = c + r ˜ ν ( f − f ) sin(2 ψ )2 α ′ ν ψ ˜ f , (5.49)where c = ˜ νν ψ sin( ϕ + ψ ) , (5.50)and a RR four form C = B ∧ C + (cid:18) c + r ˜ ν ( f − f )( ν sin ϕ cos ψ − ν cos ϕ sin ψ ) α ′ ν ψ ˜ f (cid:19) e ∧ e ∧ e ∧ e . (5.51)The non-vanishing charges of the background are the D5 and D1 charges, which take thesame values as in equations (5.16) and (5.17) Near-horizon geometry.
In the near-horizon limit the AdS and S radii, the dilatonand the T volume are given by R = α ′ ν ψ √ ν ν ˜ ν = α ′ e Φ ˜ N , e Φ = √ ν ν ν ψ ˜ ν , vol(T ) = ν ν ˜ ν = ˜ N ˜ N . (5.52)Note that both the radius and the torus volume are independent of the modulus. TheRR scalar and four form are given by C = c , C = c e ∧ e ∧ e ∧ e , (5.53)where c is defined in (5.50). 24 Conclusions
In this paper we have determined how the energies of closed perturbative strings onAdS × S × T with zero winding and momentum on the torus depend on the 20 moduliof the string theory. We focused on backgrounds which are near-horizon limits of D1/D5-and F1/NS5-branes and found that in both cases only four of the 20 moduli have a non-trivial effect on the Green-Schwarz action, and as a result have an effect on the energiesof H (0 , strings.In the near-horizon limit of D1/D5-branes one of the consequential moduli is theclosed string coupling constant whose only effect (at small g s ) is to change the radiusof curvature R . The remaining three consequential moduli come from the self-dual partof the NSNS two-form. When B + = 0, the GS action involves couplings to the RRfive-form field strength, in addition to the three form which supports the geometry. Weshowed that this new action is TrT-dual to the original action with B + = 0, and thatclosed strings in the B + = 0 background have the same energies as strings in the B + = 0background with twisted boundary conditions (3.18). The twisting is proportional tothe amount of winding and momentum on T (as well as to B + ), and so in the caseof H (0 , strings the only change to the energies comes from a change in the radius ofcurvature (1.1).In the near-horizon limit of F1/NS5-branes, the four consequential moduli are theRR scalar C and self-dual part of the two-form C +2 . The GS action in a backgroundwith a general combination of these moduli turned on is TrT-dual to the GS action withjust C turned on. As a result, for the H (0 , spectrum we may investigate the effect ofjust the C modulus: turning on the other RR moduli is encoded into a change in theradius of curvature. The GS action depends on gauge-invariant RR field strengths F p +1 = dC p − C p − ∧ H. (6.1)As a result, a constant C in the AdS geometry supported by NSNS flux induces aRR three-form flux, and the GS action is the same as the so-called mixed flux AdS backgrounds [12–14]. The interpretation is however very different. The mixed-flux back-grounds correspond to the near-horizon limit of ( p, q )-strings and 5-branes, while thebackground we investigated in this paper is a marginal deformation of the pure NSNS-flux background, and carries only F1 and NS5 charges. When C vanishes, the actionis the GS analogue of the Neveu-Schwarz-Ramond theory considered by Maldacena andOoguri [15]. With general RR moduli turned on the radius of curvature is R = α ′ k q g s c where c = C + ( C +2 ) . (6.2)Integrable methods have provided exact-in- R results for determining the energies of H (0 , states at small g s [28,36,37]. In particular, in the near horizon limit of the D1/D5-brane theory, the exact-in- R A posteriori the S matrix More accurately, a number of so-called dressing factors are not fixed by symmetries alone, but canbe found using unitarity and crossing symmetry [38] of the theory [11, 18, 30]. In the mixed flux case,the dressing factors are currently only known at the one-loop level [39]. R H (0 , spectrum can then be obtained using Bethe Ansatz methods atsmall g s . It consists of the BMN vacuum [32], on top of which magnon-like operators actto create excited states. The multiplicities of these creation operators is determined bythe BMN spectrum [32]. Each magnon has mass m = 0 , − p , whose value is determined as a solution of the Bethe Equations [11, 18]. The energyof magnon is fixed through a p -dependent shortening condition [29, 40] E ( p ) = r m + 4 h ( R ) sin (cid:16) p (cid:17) , (6.3)and the energy of the state is given by the sum of the energies of the individual magnons. R enters the above dispersion relation through the function h ( R ), whose explicit formis not fixed by integrability, much like is the case in AdS × CP [44]. As in higher-dimensional examples of integrable holography, h determines the strength of the inte-grable interactions and at large R h ( R ) = R π + . . . . (6.4)Determining h , perhaps along the lines used in AdS × CP [45], remains an importantproblem. The analysis carried out in section 3, shows that the world-sheet theorycontinues to be integrable across the 20 dimensional moduli space. The four consequentialmoduli modify the integrable structure of the H (0 , spectrum in a minimal way: theyjust changing the value of R (1.1), which changes the value of h ( R ). Since the BEs arevalid for all values of h , we retain complete control over the spectrum across the wholemoduli space as longs as g s is small. For example, in [33], the half-BPS spectrum of thetheory was found and shown to match the supergravity results of [48]. The derivationin [33] is exact in R , and combined with the present results, proves that the half-BPSspectrum does not change as we move around moduli space, in agreement with thenon-renormalization theorem of [49].Integrable methods have also been used to find the H (0 , spectrum in mixed-flux back-grounds [12–14, 39]. However, to date it was not possible to study the pure F1/NS5near-horizon geometry, because the off-shell supersymmetry algebra is not centrally ex-tended in that case [14], and these central extensions are crucial in fixing the S matrixfrom symmetries [51]. However, this obstacle occurs only at the point in moduli spacegiven in equation (2.4). As we showed in section 4, when the RR moduli are non-zero theGS action becomes equivalent to the GS mixed-flux action, for which the central exten-sions are again non-zero. We can therefore use the exact-in- R results of the mixed-flux Incorporating wrapping effects [41] into this construction remains to be fully understood [42], butit appears likely that this will be possible [43]. Perturbative world-sheet corrections to the dispersion relation in AdS × S × T and AdS × S × S × S were calculated in [46, 47]. Semi-classical strings in such backgrounds have also been studied in [50]. It is worth noting that turning on such RR moduli is expected to desingularise the dual CFT [10]. E ( p ) = r(cid:16) m + kp π (cid:17) + 4 h ( R ) sin (cid:16) p (cid:17) , (6.5)where k is the WZW level, the AdS radius is given in equation (6.2) and at large R , wehave h ( R ) = − g s ck π + . . . . (6.6)We arrive at a nice picture: the near-horizon F1/NS5-brane worldsheet theory is in-tegrable with the strength of the integrable interactions governed by ck , and the freepoint corresponding to the GS version of the WZW model solved by [15]. By analogywith higher-dimensional holography, we may think of ck as √ λ , the analogue of the ’tHooft coupling constant, which should lead to a novel planar limit for the near-horizonF1/NS5-brane theory. We intend to return to a more detailed investigation of this in thenear future.It is worth emphasizing that both in the D1/D5 and F1/NS5 backgrounds, ourarguments show that the S matrix and hence the BEs are the same as the ones ob-tained in [19, 29] and [14] respectively. The dependence on the moduli is containedentirely within the function h that enters the dispersion relation and the definition ofthe Zhukovsky variables.Given the exact-in- R nature of the integrable holographic methods now availableacross the whole moduli space, we believe the spectrum of the H (0 , sector is an idealtool for investigating more precisely the relationship between strings on AdS and itsCFT dual. It is for example striking that the dimensions of Sym N states with zerowinding and momentum also depend only on the four moduli of the Z -twisted sector,just as our H (0 , states do. Nonetheless, as the recent findings for the WZW theory at k = 1 suggest [34], the exact relationship to the Sym N orbifold most likely needs to berevisited, and the H (0 , sector contains a wealth of non-protected information with whichto test any such conjectures. It seems plausible that the more complete holographic dualwill have to incorporate aspects of an effective Higgs branch CFT [7], where integrabilityhas been found [53].We also hope to extend the analysis carried out here to strings on AdS × S × S × S backgrounds, since these too are known to be governed by integrable world-sheet theories [27, 36, 46, 54–56]. The moduli space is much smaller here [57], and someconjectures for the CFT dual also exist [57, 58]. Acknowledgements
We would like to thanks Costas Bachas, Marcus Berg, Chris Hull, Oleg Lunin, A.W. Peet,Boris Pioline, Sanjaye Ramgoolam, Leonardo Rastelli, Rodolfo Russo and KonstantinZarembo for interesting discussions, and Riccardo Borsato, Alessandro Sfondrini andAlessandro Torrielli for the many conversations on integrability and AdS3/CFT2. Wewould like to thank GGI for hosting the workshop ”New Developments in AdS3/CFT227olography” where this work begun. B.S. thanks the CERN Theory division and HelsinkiInstitute of Physics for hospitality during parts of this project. B.S. acknowledges fundingsupport from an STFC Consolidated Grant ”Theoretical Physics at City University”ST/P000797/1. This work was supported by the ERC advanced grant No 341222.
A Supergravity
A.1 Conventions
We define Hodge duality by ∗ ( e a ∧ · · · ∧ e a k ) = 1( d − k )! ǫ a ...a k b k +1 ...b d e b k +1 ∧ · · · ∧ e b d , (A.1)where ǫ = +1 . (A.2)For AdS we use the coordinate t , z and z with metric ds = − | z | − | z | ! dt + − | z | ! | dz | , (A.3)and for S we use y , y and φ , with the metric ds = − | y | | y | ! dφ +
11 + | y | ! | dy | . (A.4)The corresponding unit volume forms are given byΩ AdS = 1 + | z | (cid:16) − | z | (cid:17) dt ∧ dz ∧ dz , Ω S = 1 − | y | (cid:16) | y | (cid:17) dy ∧ dy ∧ dφ. (A.5)It is also useful to introduce the two forms ω AdS = 12 1 (cid:16) − | z | (cid:17) dt ∧ ( z dz − z dz ) ,ω S = 12 1 (cid:16) | y | (cid:17) dφ ∧ ( y dy − y dy ) , (A.6)which are define so that locally Ω AdS = dω AdS , Ω S = dω S . (A.7)28 .2 IIB supergravity The action of IIB supergravity is given by S IIB = 12 κ Z √− g (cid:18) e − (cid:16) R + 4( ∂ Φ) − H (cid:17) − (cid:16) F + F +
12 5! F (cid:17)(cid:19) − κ Z C ∧ H ∧ F , (A.8)where the gauge invariant field strengths are given in terms of gauge potentials by H = dB, F = dC − C H, F = dC − C ∧ H,F = dC , F = dC − C ∧ H, F = dC − C ∧ H, (A.9)and satisfy ∗ F = F , ∗ F = − F , ∗ F = F . (A.10)The RR field strengths satisfy the Bianchi identities and equations of motions dF = 0 , dF + F ∧ H = 0 , dF + F ∧ H = 0 ,dF + F ∧ H = 0 , dF + F ∧ H = 0 . (A.11)The Bianchi identity and equation of motion for the NSNS three form H are given by dH = 0 , d (cid:16) e − ∗ H (cid:17) − F ∧ ∗ F − F ∧ F = 0 . (A.12)The equations of motion for the metric g µν are given by R ab − ∇ a ∂ b Φ + 12 | H | ab + 12 e (cid:18) | F | ab + | F | ab + | F | ab − η ab (cid:16) | F | + | F | (cid:17)(cid:19) = 0 , (A.13)where the symmetric contractions of the fields strengths are given, for example, by | H | ab = h i a H, i b H i = − ∗ ( i a H ∧ ∗ i b H ) . (A.14)The equation of motion of the dilaton reads d ∗ d Φ − | d Φ | + | H | − e (cid:16) | F | + | F | (cid:17) = 0 . (A.15)From the equations of motion and Bianchi identities we define the conserved chargescarried by D p and NS5 branes as well as F1 strings. For example, the D3 charges isgiven by Q D3 = 12 κ Z M ⊥ dF + F ∧ H, (A.16)where M ⊥ is transverse to the branes. This charge is local and quantised, but is onlyinvariant under small gauge transformations [20]. By partially integrating this in theradial direction away from the branes, this can be rewritten as an integral over a surface ∂ M ⊥ enclosing the branes. The exact form of the integrand will then depend on theexact form of the involved field strengths. We follow the conventions of [59]. Note that the Bianchi identity for F n is the same as the equation of motion for ∗ F n . .3 T duality If we make a T-duality transformation along the direction z then the NSNS fields trans-form as ˜ B µz = − g zµ g zz , ˜ B µν = B µν + 2 g zz B z [ µ g ν ] z , ˜Φ = Φ −
12 log g zz , ˜ g zz = 1 g zz , ˜ g µz = − B µz g zz , ˜ g µν = g µν − g zz (cid:16) g µz g νz − B µz B νz (cid:17) , (A.17)and the RR fields transform as˜ C ( n ) µ ··· µ n − z = C ( n − µ ··· µ n − − ( − n n − g zz C ( n − z [ µ ··· µ n − g µ n − ] z , ˜ C ( n ) µ ··· µ n = C ( n +1) µ ··· µ n z − nC ( n − µ ··· µ n − B µ n ] z − ( − n n ( n − g zz C ( n − z [ µ ··· µ n − B µ n − | z | g µ n ] z , (A.18)where we have used a superscript to indicate the degree of the various forms. B General AdS × S × T backgrounds In this appendix we will write down two general AdS × S × T type IIB supergravitybackgrounds, with the metric given by ds = R (cid:16) ds + ds S (cid:17) + g ij dx i dx j . (B.1)We will focus on two special cases: backgrounds carrying only the RR charges of theD1/D5 system, and hence supported by a RR three form F and its dual F , and back-grounds carrying F1 and NS5 charges, and thus sourcing the NSNS field strength H , andits dual.There are a number of fields we can turn on that are compatible with the isometriesof the above metric: • a constant B field on T , • a constant two-form potential C on T , • the RR scalar C , • a constant dilaton Φ, • a RR four-form potential C on T .In the following two subsections we will write down general solutions with these fieldsturned on. B.1 RR backgrounds
Let us start by considering the case of backgrounds supported by RR flux. Imposingthe equations of motion, as well as demanding the absence of D3 charges as well as any30SNS charges, we find that the dilaton is an arbitrary constant, that the constant Bfield is self dual B = b (cid:16) e ∧ e + e ∧ e (cid:17) + b (cid:16) e ∧ e − e ∧ e (cid:17) + b (cid:16) e ∧ e + e ∧ e (cid:17) , (B.2)and that the RR potentials are given by C = c , C = 0 ,C = f (cid:16) ω AdS + ω S (cid:17) + c ij e i ∧ e j C = f (cid:16) ω AdS + ω S (cid:17) ∧ B + C ∧ B + c e ∧ e ∧ e ∧ e ,C = − f (cid:16) ω AdS + ω S (cid:17) ∧ e ∧ e ∧ e ∧ e , (B.3)with c ji = − c ij . This corresponds to the field strengths F = 0 , F = 0 F = f (cid:16) Ω AdS + Ω S (cid:17) , F = f (cid:16) Ω AdS + Ω S (cid:17) ∧ B,F = − f (cid:16) Ω AdS + Ω S (cid:17) ∧ e ∧ e ∧ e ∧ e . (B.4)This results in a solution to the IIB supergravity equations of motion, provided the radiusof AdS and S is related to the other parameters by R = e f (cid:16) b + b + b (cid:17) . (B.5)This solution includes a number of free parameters: • the metric of T – 10 parameters, • the two-form potential C on T – 6 parameters, • the self-dual B field – 3 parameters, • the dilaton Φ – 1 parameter, • one linear combination of the RR potentials C and C – 1 parameter, • the coefficient f of the RR three-form field strength F – 1 parameter.The last parameter, the coefficient f in front of F , gives the radius of AdS and S .This leaves us with 21 parameters – one more than the expected number of moduli inthe D1/D5 system. The reason for this is that the total volume of T is fixed by theattractor mechanism when the AdS × S × T geometry is obtained in the near-horizonlimit of the brane system. To find the volume of T in directly in the above supergravitysolution, we need to impose flux quantisation of the field strengths F and F . This givesus two additional constraints. One such constraint fixes f , and hence the radius R , interms of the fluxes, and the second constraint determines the volume of T .Note that of the 21 remaining parameters, only the self-dual B field appears in theexpressions for the field strengths. The Green-Schwarz string does not couple directly tothe gauge potentials, but only to the gauge invariant field strengths. Hence, the onlymoduli the closed string spectrum in the D1/D5 system is sensitive to originates in theself-dual B field. From (A.9) we see that for H = 0 the gauge invariant field strengths only depend on the derivativesof the gauge potentials. .2 NSNS backgrounds Let us now consider backgrounds that only carry F1 and NS5 charges. Again we imposethe equations of motion and Bianchi identities, as well as the vanishing of all D-branecharges. We then find that the dilaton is constant and that the RR potentials are givenby C = c , C = − c e ∧ e ∧ e ∧ e ,C = c ( e ∧ e + e ∧ e ) + c ( e ∧ e − e ∧ e ) + c ( e ∧ e + e ∧ e ) , (B.6)while the B field is given by B = h (cid:16) ω AdS + ω S (cid:17) + b ij e i ∧ e j , (B.7)with b ji = − b ij . The corresponding field strengths are given by H = h (Ω AdS + Ω S ) , (B.8)and F = 0 , F = 0 ,F = − c h (Ω AdS + Ω S ) ,F = − h (Ω AdS + Ω S ) ∧ C ,F = + c h (Ω AdS + Ω S ) ∧ e ∧ e ∧ e ∧ e . (B.9)The radius of AdS and S is now given by R = h (cid:16) e ( c + c + c + c ) (cid:17) . (B.10)Again, this solution has 22 free parameters: • the metric of T – 10 parameters, • the self-dual two-form potential C – 3 parameters, • the B field – 6 parameters, • the dilaton Φ – 1 parameter, • one linear combination of the RR potentials C and C – 1 parameter, • the coefficient h of the NSNS three-form field strength H – 1 parameter.As in the previous case, imposing charge quantisation lets us fix the coefficient h , andhence the radius, as well as the volume of T in terms of the brane charges of thebackground. We are then left with the 20 expected moduli. C The TrT transformed Green-Schwarz string
In this appendix we write down the expressions for the terms in the gauge-fixed Green-Schwarz string after two TrT transformations with parameters ϕ and ψ . In general, theLagrangian can be written as L = − (cid:18) γ αβ ∂ α x i ∂ β x j G ij − ǫ αβ ∂ α x i ∂ β x j B ij + 2 ∂ α x i (cid:16) γ αβ U β,i − ǫ αβ V β,i (cid:17) + L rest (cid:19) . (C.1)32e will work to quadratic order in fermions and decompose the coefficients in the La-grangian as G ij = G ( b ) ij + G ( f ) ij , B ij = B ( b ) ij + B ( f ) ij , U α,i = U ( b ) α,i + U ( f ) α,i , V α,i = V ( b ) α,i + V ( f ) α,i , (C.2)where, e.g. , G ( b ) ij is purely bosonic while G ( f ) ij is quadratic in fermions. For concreteness wewill assume that there are four U(1) directions, which are parametrised with x , . . . , x .We further set the volume of these four directions to be equal and assume that there isno B field and that the metric is diagonal. The bosonic background fields then take theform G ( b ) ij = δ ij s µ µ , B ( b ) ij = 0 , U ( b ) α,i = 0 , V ( b ) α,i = 0 . (C.3)Furthermore, we set G ( f ) ij = δ ij g f s µ µ , B ( f )89 = B ( f )67 , B ( f )79 = − B ( f )68 , B ( f )78 = B ( f )69 . (C.4)Performing the two TrT transformations and keeping only terms that are at mostquadratic in fermions, we find that the bosonic part of the resulting background is givenby ˜ G ( b )66 = ˜ G ( b )77 = √ µ µ µ ϕ , ˜ B ( b )67 = µ − µ µ ϕ sin ϕ cos ϕ, ˜ G ( b )88 = ˜ G ( b )99 = √ µ µ µ ψ , ˜ B ( b )89 = µ − µ µ ψ sin ψ cos ψ. (C.5)33he terms in the transformed background that are quadratic in fermions are given by˜ G ( f )66 = ˜ G ( f )77 = √ µ µ µ ϕ (cid:16) ( µ cos ϕ − µ sin ϕ ) g f − µ cos ϕ sin ϕ B ( f )67 (cid:17) , ˜ G ( f )88 = ˜ G ( f )99 = √ µ µ µ ψ (cid:16) ( µ cos ψ − µ sin ψ ) g f − µ cos ψ sin ψ B ( f )67 (cid:17) , − ˜ G ( f )79 = ˜ G ( f )68 = √ µ µ µ ϕ µ ψ µ sin( ϕ − ψ ) B ( f )69 , ˜ G ( f )78 = ˜ G ( f )69 = √ µ µ µ ϕ µ ψ µ sin( ϕ − ψ ) B ( f )68 , ˜ B ( f )67 = µ µ ϕ (cid:16) ( µ cos ϕ − µ sin ϕ ) B ( f )67 + 2 µ cos ϕ sin ϕ g f (cid:17) ˜ B ( f )89 = µ µ ψ (cid:16) ( µ cos ψ − µ sin ψ ) B ( f )67 + 2 µ cos ψ sin ψ g f (cid:17) , − ˜ B ( f )79 = ˜ B ( f )68 = µ µ ϕ µ ψ ( µ cos ϕ cos ψ + µ sin ϕ sin ψ ) B ( f )68 , ˜ B ( f )78 = ˜ B ( f )69 = µ µ ϕ µ ψ ( µ cos ϕ cos ψ + µ sin ϕ sin ψ ) B ( f )69 , ˜ U ( f ) α, = µ µ ϕ cos ϕ U ( f ) α, + √ µ µ µ ϕ sin ϕ V ( f ) α, , ˜ U ( f ) α, = µ µ ϕ cos ϕ U ( f ) α, − √ µ µ µ ϕ sin ϕ V ( f ) α, , ˜ U ( f ) α, = µ µ ψ cos ψ U ( f ) α, + √ µ µ µ ψ sin ψ V ( f ) α, , ˜ U ( f ) α, = µ µ ψ cos ψ U ( f ) α, − √ µ µ µ ψ sin ψ V ( f ) α, , ˜ V ( f ) α, = µ µ ϕ cos ϕ V ( f ) α, + √ µ µ µ ϕ sin ϕ U ( f ) α, , ˜ V ( f ) α, = µ µ ϕ cos ϕ V ( f ) α, − √ µ µ µ ϕ sin ϕ U ( f ) α, , ˜ V ( f ) α, = µ µ ψ cos ψ V ( f ) α, + √ µ µ µ ψ sin ψ U ( f ) α, , ˜ V ( f ) α, = µ µ ψ cos ψ V ( f ) α, − √ µ µ µ ψ sin ψ U ( f ) α, . (C.6) References [1] J. M. Maldacena, “The large N limit of superconformal field theories and supergravity” , Adv. Theor. Math. Phys. 2, 231 (1998) , hep-th/9711200 .[2] A. Strominger and C. Vafa, “Microscopic origin of the Bekenstein-Hawking entropy” , Phys. Lett. B379, 99 (1996) , hep-th/9601029 .
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