Heating up the Baryonic Branch with U-duality: a unified picture of conifold black holes
MMCTP-11-01UTTG-01-11
Heating up the Baryonic Branch with U-duality:a unified picture of conifold black holes
Elena C´aceres ∗ , Carlos N´u˜nez † and Leopoldo A. Pando Zayas ∗∗ ∗ Facultad de CienciasUniversidad de ColimaBernal Diaz del Castillo 340, Colima, M´exico. andTheory Group, Department of Physics,University of Texas at Austin, Austin, TX 78727, USA. † Department of PhysicsUniversity of Swansea, Singleton ParkSwansea SA2 8PPUnited Kingdom. ∗∗ Michigan Center for Theoretical PhysicsRandall Laboratory of Physics, the University of MichiganAnn Arbor, MI 48109-1040. USA
Abstract
We study different aspects of a U-duality recently presented by Maldacena and Martelliand apply it to non-extremal backgrounds. In particular, starting from new non-extremalwrapped D5 branes we generate new non-extremal generalizations of the Baryonic Branch ofthe Klebanov-Strassler solution. We also elaborate on different conceptual aspects of theseU-dualities, like its action on (extremal and non-extremal) Dp branes, dual models for Yang-Mills-like theories, generic asymptotics and decoupling limit of the generated solutions. [email protected] [email protected] [email protected] a r X i v : . [ h e p - t h ] F e b ontents N = 1 SUSY Solutions 9 N = 1 Black Holes . . . . . . . . . . . . . . . . . . . 20 Introduction
The Maldacena conjecture [1] and some of its refinements [2] are the guiding principle be-hind much of the progress in the interface String Theory-Quantum Field theory in the lasttwelve years. The influence of this approach extends to toy models with different numberof supersymmetries (SUSY’s), systems at finite temperature and/or finite density and lowerdimensional systems. The applications to physically relevant systems do not seem to beexhausted and on the contrary, increase with time. In this sense, finding new (trustable)solutions to the equations of motion of String Theory (even in the point particle and classicalapproximation) has become an industry with various applications in Physics and Mathematics.Solution generating techniques have certainly played a role. Some examples worth mentioningare the combination of T-dualities and shifts of coordinates that generated solutions dual tominimally SUSY superconformal field theories (beta deformations) or duals to non-relativisticfield theories [3] .In this paper we will focus our attention on a particular solution generating technique thatwas presented in [6]. The procedure suggested by these authors starts by taking a solution intype IIB string theory proposed to be dual to (a suitably UV-completed version of) N = 1Super Yang-Mills in four dimensions. The solution has the topology R , × M where M preserves four supercharges. The algorithm to generate the new solutions goes as follows: firstapply a set of three T-dualities in the R directions, which leaves us with a IIA configuration;lift this configuration to M-theory and boost–with rapidity β –in the eleventh direction; thenreduce to IIA and T-dualize back in R . This generates a solution that roughly speaking isthe dual description to the baryonic branch [7] of the Klebanov-Strassler field theory [8]. Forall the technical details of this procedure the reader can refer to [6] or to our Appendix E.The ‘seed solution’ (as we will refer to the initial solution on which the generating algorithmis applied) was discussed in the papers [9] (Section 8) and [10] (Section 4.3). In a particularlimit, this becomes the exact solution discussed in [11].We will also refer to the solution generating algorithm, that is a U-duality, as ‘rotation’in a sense explained below and in previous papers. In the following we will emphasize someaspects of the generating technique presented in [6] that we find particularly interesting andhave not been explicitly discussed in the existing bibliography: • In the case of [6], the rotation generates D3 brane charge. This is the reason why the SO (1 ,
3) isometry of the background is untouched, in spite of doing different operationsin time (boost) and in the R directions (T-dualities). As a by-product of the generationof D3 branes the dilaton is invariant under this set of operations. In other areas of Physics, solution generating techniques play a major role. For example, many of theinteresting solutions in higher dimensional gravity were constructed using solution generating algorithms. Seefor example [4] for original papers and [5] for a nice summary with an important application. It can be seen that this U-duality (sequence of T-dualities, lifts and boosts) describedabove is equivalent to a particular case of ‘rotation’ in the SU (3) structure of the mani-fold M which is characterized by a complex 3-form Ω and a 2-form J - see the papers[12] and [13]. This is another way of understanding the presence of the SO (1 ,
3) isometry:from this perspective, all the rotation ‘occurs’ in the internal M . • The rotation generates a solution that contains two free parameters, the value of thedilaton at infinity (already present in the seed solution) and the boost parameter β .In order for the final background to be dual to the Klebanov-Strassler QFT decoupled from gravity, we must take the limit β → ∞ so that the generated warp factor vanishesasymptotically. In this sense, the seed solution describes a field theory coupled to gravity.Only in a particular limit-see the discussion in [6], [13], we approach the near branesolution of [11]. • This rotation or U-duality, is a very curious operation from the viewpoint of the dual fieldtheory. Indeed, it generates (aside from the bifundamental fields) new global symmetries,like the baryonic U (1). • In the same vein: the connection between the two field theories from a geometric view-point (Field theory at strong coupling!) is just the mentioned U-duality starting fromthe D • After the first set of T-dualities described above, we have a IIA background with D2brane charges. When lifted to M-theory we have M2 branes, that after boosted generateM2 charge and an M-wave. When reduced, this generates a D0, D2 and NS H field.The generation of the H in the presence of RR three-form implies the need to generate F . See Appendix E for technical details. • We emphasized that the U-duality ( chain of T-dualities, lifts and boost) described aboveis a particular case of the rotation of J , Ω discussed in [12], [13], [14]. On the otherhand, the rotation of J , Ω relies on the background being supersymmetric, while theU-duality can be also applied to Non-SUSY backgrounds. This will play an importantrole in the rest of this paper. In this paper we will try to gain a different perspective on this solution generating technique.We will not focus on the rotation of J, Ω but rely on the U-duality description (in spite of the3atter being a particular case of the former). Our interests will be two-fold. On one hand, wewill try to get a better handle on the generating algorithm by changing it (suggesting otherrelated algorithms) and applying it to various cases. On the other hand, we will apply itto backgrounds that do not preserve SUSY. We will study the effect of the rotation on blackhole solutions, generating new non-extremal solutions with horizons and other curious featuresthat will be discussed. We will also start the study of the properties of those newly generatedsolutions.This paper is organized as follows: In sections 2 and 2.1 as a warm-up exercise we discussa chain of dualities and boost acting on non-extremal Dp branes. In section 2.1.1 we willapply this to rotate a background dual to a version of Yang-Mills. Then, in section 2.2 we willstudy a variation of this U-duality that will clarify various aspects of the coming material. InSection 3 we will U-dualize a new solution describing the non-extremal deformation of a stackof N c D5 branes wrapping a two-cycle inside the resolved conifold. We will elaborate uponvarious aspects of this particular rotation in section 3.5. Finally in Section 4 we comment ondecoupling aspects of the rotation. Various appendices complement our presentation. Theyhave been written with plenty of detail hoping that colleagues working on these topics willfind them useful. We close with a summary, conclusions and a list of possible future projects.
In this section, we will start with a simple example of the ‘rotation’ procedure. That is, asequence of T-dualities, bringing the background to a type IIA solution of the supergravityequations of motion, followed by a lift to eleven dimensions, where we will boost the config-uration. We will then reduce to IIA and T-dualize back, to what we will call the ‘rotatedbackground’. As a toy example, in this section we will rotate flat Dp branes; first in p direc-tions and then in p − q directions. We will do this in detail to appreciate the differences thisintroduces in the generated solution. In Appendix D, we will propose and analyze anotherpossible U-duality to generate new solutions. Consider backgrounds of IIA/IIB of the form, ds = H ( ρ ) − / h − h ( ρ ) dt + d~x p i + H ( ρ ) / h dρ h ( ρ ) + ρ d Ω − p i ,e φ [ initial ] = e φ ( ∞ ) H − p , ∗ F p +2 = Q Vol Ω − p ,F p +2 = − ∂ ρ A ( ρ ) dt ∧ dx ∧ .... ∧ dx p ∧ dρ (2.1)4ith, H ( ρ ) = 1 + (cid:16) L p ρ (cid:17) − p , h ( ρ ) = 1 − (cid:16) R T ρ (cid:17) − p , A ( ρ ) = αg s H ( ρ ) , (2.2)where α = ˜ QL − pp and Q = (7 − p ) ˜ Q is related to the charge of the solution (see Appendix B fordetails). If we choose R T = 0, the configuration above is typically singular (except for p = 3)and preserves 16 SUSYs. Recall that for even (odd) values of p , we are dealing with a solutionof Type IIA(B) supergravity.Now, we will ‘rotate’ this background, by first T-dualizing in the p directions, this will bringus to a IIA solution, we will lift the solution to eleven-dimensional supergravity and performa boost of rapidity β in the eleventh-direction. We will then reduce to IIA and T-dualizeback in the ~x p coordinates to obtain what we call the ‘rotated background’. Our final rotatedbackground is given by , ds = H ( ρ ) − / R ( ρ ) h − h ( ρ ) dt + d~x p i + R ( ρ ) H ( ρ ) / h dρ h ( ρ ) + ρ d Ω − p i (2.3) e φ [ final ] = e φ [ initial ] R ( ρ ) − p = e φ ( ∞ ) ( H ( ρ ) R ( ρ ) ) − p (2.4) F p +2 = − ∂ ρ a ( ρ ) dt ∧ dx ∧ .... ∧ dx p ∧ dρ (2.5)where we have defined, R = ( A ( ρ ) sinh β + cosh β ) − h ( ρ ) sinh βg s H ( ρ ) , (2.6) a = 1 R [ A ( ρ ) cosh 2 β + ( A ( ρ ) + 1) cosh β sinh β − h ( ρ )2 g s H ( ρ ) sinh(2 β )] (2.7)We have explicitly checked that the rotated background is a solution of the equations ofmotion. Notice that eq. (2.6) can be written as R = cosh β + A ( ρ ) sinh 2 β + ( R T L p ) − p sinh βH ( ρ ) g s (2.8)which makes clear that R is strictly positive. In a similar way, eq. (2.7) can be written as a = 1 HR " g s α cosh(2 β ) + (cid:18) R T L p (cid:19) − p sinh β cosh βg s + H ( ρ ) sinh β cosh β . (2.9)The metric (2.3) has the structure of a warped space, ds = H ( ρ ) − / h − h ( ρ ) dt + d~x p i + H ( ρ ) / h dρ h ( ρ ) + ρ d Ω − p i , (2.10) In Appendix A, we will present the intermediate steps of this calculation for the extremal case h = 1. H ( ρ ) = H ( ρ ) R ( ρ ) = H ( ρ ) cosh β + αg s sinh 2 β + ( R T L p ) − p sinh βg s (2.11)Note that H ( ρ ) is a harmonic function of the transverse space. Hence, after the rotation in p directions we are left with a Dp brane solution. Indeed, in terms of the new warp factor, thedilaton and gauge potential are e φ [ final ] = e φ ( ∞ ) H ( ρ ) − p ∂ ρ a ( ρ ) = (cid:20) αg s + ( R T L p ) − p tanh βg s (cid:21) ∂ ρ (cid:16) H ( ρ ) (cid:17) = ˜ α ∂ ρ (cid:16) H ( ρ ) (cid:17) , (2.12)which together with (2.10) define a Dp brane background with a β dependent RR charge. Atinfinity the warp factor asymptotes to a constant, H ( ρ ) ∼ cosh β + αg s sinh 2 β + ( R T L p ) − p sinh βg s , (2.13)and the space is asymptotically flat, as expected. However, it is interesting to note that evenif we start with the Dp branes after the decoupling limit is taken, that is H = L − pp ρ − p , we willhave in the UV that, H ( ρ ) ∼ L − pp ρ − p cosh β + αg s sinh 2 β + ( R T L p ) − p sinh βg s (2.14)So, again, the warp factor asymptotes to a constant. In other words, this rotation is takingthe configuration out of the decoupling limit or coupling the field theory modes to gravity.As is well known, before the rotation, the charge of the Dp brane solution is quantized.After the rotation the charge is,1(2 πl s ) − p Z S − p ∗ F p +2 = ˜ α ρ − p H ( ρ ) V ol S − p = Q cosh β + R T − p c p g s sinh 2 β (2.15)where c p = (7 − p ) V ol S − p (2 πl s ) − p g s . Note that, generically, the charge is not quantized. This is notunusual since the supergravity dualities involved in the rotation procedure are a symmetry ofthe supergravity equations and not of the full string theory.We will now move to study a more interesting example from the viewpoint of gauge-stringsduality. We will apply the rotation procedure to the dual of a non-SUSY Yang-Mills-liketheory, first presented in [17] and further studied in [18], [19].6 .1.1 Example: rotation of a dual to a Yang-Mills-like theory The original background, consists of the decoupling limit of a stack of N c D4 branes wrappinga circle with SUSY breaking boundary conditions. It was discussed with details in variouspublication, see for example [17], [18], [19]. We summarize it here (in string frame) ds = H ( ρ ) − / h − h ( ρ ) dt + d~x + f ( ρ ) dx i + H ( ρ ) / h dρ f ( ρ ) h ( ρ ) + ρ d Ω i ,e φ [ initial ] = g s H − ,F = − ∂ ρ A ( ρ ) dt ∧ dx ∧ .... ∧ dx ∧ dρ F = ∗ F (2.16)where A ( ρ ) = g s H ( ρ ) and V ol is the volume of the unit four- sphere Ω .. The functions (inthe low/zero temperature phase) are given by H = ( Lρ ) , f = 1 − ( R kk ρ ) , h = 1 , (2.17)where L = N c πg s l s and the coefficient R kk is a free parameter. In the high temperaturephase (with the coordinate x compactified) we have, H = ( Lρ ) , h = 1 − ( R T ρ ) , f = 1 (2.18)where R T is a free parameter related to the temperature of the system, T = 34 π ( L R T ) − / (2.19)Proceeding as described in the previous section we rotate the background by applying the U-duality already discussed: four T-dualities in the worldvolume coordinates, uplift to M-theory,boost, reduce to IIA and T dualize back. The rotated background is, ds = H ( ρ ) − / h − h ( ρ ) dt + d~x + f ( ρ ) dx i + H ( ρ ) / h dρ f ( ρ ) h ( ρ ) + ρ d Ω i ,e φ [ final ] = g s H − ,F = a ( ρ ) dt ∧ dx ∧ .... ∧ dx ∧ dρ (2.20)where the functions H ( ρ ) , a ( ρ ) now read H ( ρ ) = H ( ρ ) cosh β + 1 g s sinh 2 β + ( R T L ) sinh βg s (2.21) a ( ρ ) = 1 R ( ρ ) h A ( ρ ) cosh 2 β + ( A ( ρ ) + 1) cosh β sinh β − f ( ρ ) h ( ρ ) sinh(2 β )2 g s H ( ρ ) i (2.22) In [18] the authors performed a rescaling of the RR potential: C p +1 → κ µ − p π C p +1 and as a result theRR charge in their paper is measured in units of 2 π . We do not perform such rescaling here. R ( ρ ) = H ( ρ ) H ( ρ ) . As before, one can check explicitly that the background above is a solutionof the eqs. of motion, but for this we must set either R T = 0 for the low/zero temperaturephase or set R kk = 0, compactifying the x direction, in the high temperature phase.As explained in the previous section, in spite of having started with a decoupled background,after the rotation we have an asymptotically flat space, indicating that the rotation has coupledthe field theory modes to the gravitational ones. We will study now a peculiar situation, where we have Dp branes and we separate a q manifoldfrom the p + 1 dimensional world-volume (we focus in the case q = 2 here, the reason for thiswill be explained in Appendix C). The internal manifold need not be specified. We will imposeon our U-duality that: • after the initial T-dualities, we are in IIA background, so that this is upliftable to M-theory • that the boost generates a NS H field, or what is equivalent, that the IIA configurationafter the initial T-dualities contains an electric F .We will assume (though this need not be the case) that the internal q-manifold is a torus. Inthat case, the solution reads (in string frame as usual) , ds = H − / [ − dt + dx p − q + dσ q ] + H / [ dρ + ρ d Ω − p ] ,F p +2 = ∂ ρ ˆ Adt ∧ dx ∧ ... ∧ dx p − q ∧ dσ q ∧ dρ,e φ [ initial ] = e φ ( ∞ ) H − p . (2.23)We will proceed as follows: first we will do T dualities in the x p − q directions, this will leaveus in a IIA configuration, we will lift then to M-theory and boost. We reduce to IIA and Tdualize back, all the details are written in Appendix C. We end up this duality-chain with afinal configuration ds II,st = g tt dt + d~x p − H / B / + B / (cid:16) H − / d~σ + H / [ dρ + ρ d Ω − p ] (cid:17) ,F p +2 = ∂ ρ ˆ A (cosh β + a t sinh β ) ∧ dρ ∧ dσ ∧ dσ ∧ dx ∧ .... ∧ dx p − ,H = sinh β∂ ρ ˆ Adρ ∧ dσ ∧ dσ ,F p = ∂ ρ ( a t ) dt ∧ dρ ∧ dx ∧ .... ∧ dx p − ,e φ [ final ] = B − p H − p (2.24) we are dealing here with the Lorentz invariant case. Working with a non-extremal factor is straightforward. A = H − / [ H sinh β − cosh β ] , B = H − / [ H cosh β − sinh β ] ,C = 2 H − / sinh β cosh β (1 − H ) , g tt = 4 AB − C √ B ,a t = C B = sinh β cosh β (1 − H ) H cosh β − sinh β , (2.25)Various aspects are worth emphasizing in this final background. Notice that: • the metric is dual to a field theory with SO (1 , p −
2) isometry, since(4 AB − C ) H / = − → g tt = − g x i x i (2.26) • for p = 5 the dilaton is invariant under the whole U-duality procedure, since e φ [ initial ] = H − p = e φ [ final ] = B − p H − p (2.27) • for p = 5 the procedure is generating charge of D3 brane (represented by the F p =5 ) wherethe D3 branes are extended in the p − q directions. This is the reason why the dilatondoes not change as the D3 branes do not couple to it. The field H is also generated.In following sections, we will study a particular case of this U-duality, for a situation in whicha set of D5 branes wrap a curved two-manifold (but we also have a black hole in the metric,breaking the SO (1 ,
3) isometry), the results are qualitatively the same. One should emphasizethat there is yet another way of generating NS three-form fields, that is basically starting fromNS five branes in IIA wrapping a three-cycle, see [20] for details.Also, notice that we have imposed that after the first set of T-dualities, the backgroundis solution of IIA Supergravity (the conditions for this to happen are discussed in AppendixC). Were this not the case, we present in Appendix D another possible solution generatingtechnique with some applications.Now, we will move to study an interesting application of the U-dualities discussed above. N = 1 SUSY Solutions
As anticipated, in this section we will apply the chain of dualities, lift and boost proposedin [6], to generate a new solution starting from a non-extremal solution in Type IIB. Theinterest of the original (‘seed’) solution is that it was argued to be dual to a field theory withminimal SUSY in four dimensions. The field theory was studied (at weak coupling) in [16] andis basically N = 1 Super-Yang-Mills plus a sets of (massive) KK vector and chiral superfields9hat UV complete the dynamics. This system was well studied and various string duals areknown (particular solutions describing the field theory at strong coupling, with VEV’s andcertain operators deforming the Lagrangian). Let us briefly describe the general form ofthe string dual. The ‘seed’ background describes the backreaction of a set of N c D5 braneswrapping a two-cycle inside the resolved conifold. It consists of a metric, dilaton φ ( ρ ) and RRthree-form F and, in string frame, it reads (the coordinates used are [ t, ~x, ρ, θ, ϕ, ˜ θ, ˜ ϕ, ψ ]), ds = e φ ( ρ ) h dx , + e k ( ρ ) dρ + e h ( ρ ) ( dθ + sin θdϕ ) ++ e g ( ρ ) (cid:0) (˜ ω + a ( ρ ) dθ ) + (˜ ω − a ( ρ ) sin θdϕ ) (cid:1) + e k ( ρ ) ω + cos θdϕ ) i ,F (3) = N c " − (˜ ω + b ( ρ ) dθ ) ∧ (˜ ω − b ( ρ ) sin θdϕ ) ∧ (˜ ω + cos θdϕ ) + b dρ ∧ ( − dθ ∧ ˜ ω + sin θdϕ ∧ ˜ ω ) + (1 − b ( ρ ) ) sin θdθ ∧ dϕ ∧ ˜ ω . (3.28)where ˜ ω i are the left-invariant forms of SU (2)˜ ω = cos ψd ˜ θ + sin ψ sin ˜ θd ˜ ϕ, ˜ ω = − sin ψd ˜ θ + cos ψ sin ˜ θd ˜ ϕ , ˜ ω = dψ + cos ˜ θd ˜ ϕ. (3.29)This supersymmetric system was carefully studied in a series of papers [9], [10]; where it wasshown that there is a combination of background functions (basically a ‘change of basis’) thatmove from a set of coupled BPS equations to a decoupled one that one can solve-up to onefunction, that we will call P ( ρ ). We will not insist much with this formalism here and referthe interested reader to the original work [9], [10] .We will be more restrictive and for the purposes of this paper, we will study solutions wherethe functions a ( ρ ) = b ( ρ ) = 0 in the background of eq.(3.28). This is just in order to makeour numerics simpler and illustrate the points we want to make. If we wanted to constructa black hole solution showing explicitly the transition between R-symmetry breaking and itsrestoration, we should then work without this restriction. Since in all known solutions theasymptotics of the functions a ( ρ ) ∼ b ( ρ ) ∼ e − ρ our asymptotic results will be qualitativelycorrect, but near the black hole horizon there could be important differences.We will then proceed as follows: first we will propose a background including a black holewith the restrictions mentioned above ( a ( ρ ) = b ( ρ ) = 0). We will then pass it through thesolution generating machine. This will produce a new background, now including also F , H ,that we will write explicitly (we have checked that the equations of motion before and afterthe U-duality are the same). We will be explicit about the asymptotics of each of the functions All this formalism was also applied to the case in which one also adds fundamental matter to the dualQFT or sources to the background above. We refer the interested reader to the review [21]. Z N → Z . In this section we consider the background presented in equation (3.28) with the restrictions a = b = 0 but including the non-extremality factors. In string frame we have, ds IIB,s = e φ h − h ( ρ ) dt + dx + dx + dx i + ds ,s ,ds = e φ h e k s ( ρ ) dρ + e k ω + cos θdϕ ) + e q ( dθ + sin θdϕ ) + e g d ˜ θ + sin ˜ θd ˜ ϕ ) i ,F (3) = N c " − ˜ ω ∧ ˜ ω + sin θdθ ∧ dϕ ∧ (˜ ω + cos θdϕ ) = α N c w , (3.30)where h ( ρ ) , s ( ρ ) are the non-extremality functions. In the following, we will set s ( ρ ) = h ( ρ )which is simply a choice of parametrization; this implies that (on a particular solution) otherbackground functions { q ( ρ ) , g ( ρ ) , k ( ρ ) , φ ( ρ ) } need not take the same values as in the SUSYbackground.The above Ansatz might be familiar to some readers. It is worth highlighting a key differencewith previous work. Since our goal is to ‘rotate’ this solution we look for solutions withstabilized dilaton. Namely, the typical asymptotic value of the dilaton inherited from the D5(or NS5 brane) solution is linear [32], more precisely [23] [25], e φ ∼ e ρ ρ − / (3.31)We are looking for a dilaton that stabilizes at infinity, that is, which asymptotically behavesas: e φ ∼ O (cid:0) e − / ρ (cid:1) . (3.32)Our solution is qualitatively characterized by two parameters: one describing the nonextremal-ity of the solution and the other the speed at which the dilaton gets stabilized.After applying the solution generating technique, we end up with a new Type IIB back-ground containing F , H aside from the fields originally present in eq.(3.30) (all details are11ritten in the Appendix E), ds IIB,st = H − / h − hdt + dx + dx + dx i + e φ H / h e k dρ h ( ρ ) + e k ω + cos θdϕ ) ++ e q ( dθ + sin θdϕ ) + e g d ˜ θ + sin ˜ θd ˜ ϕ ) i ,H / = cosh β q e − φ − h ( ρ ) tanh β, φ new = φ ( ρ ) ,F = ( α N c βw , H = −√ h ( α N c β e φ ∗ w F = − tanh β (1 + ∗ )vol ∧ d ( hH ) . (3.33)In the following, we will focus in the limit β → ∞ , this is the field theory-limit where thewarp factors vanish at infinity. We will then rescale˜ N = N cosh β, x i → p cosh β p ˜ N α x. (3.34)With all the above the β → ∞ limits are finite and the solution is given by ds IIB,st = ˜
N α h ˜ H − / ( − hdt + dx i dx i ) + e φ ˜ H / ds i ,F = α ˜ N w , H = −√ h α ˜ N e φ ∗ w F = − ( α ˜ N ) (1 + ∗ )vol ∧ d ( h ˜ H ) (3.35)with ˜ H / = p e − φ − h ( ρ ) (3.36)It is worth making a few observations: • We have two new classes of solutions. The equations of motion before and after the‘rotation’ are the same, thus a solution of the system of equations (3.30) furnishes asolution of the form (3.33)-(3.35). • The non-extremality factor have made its way into the RR five form and the warp factorin eq.(3.36), but the dilaton and the three forms are unchanged from the extremal case.Notice that the factor of √ h in the NS field H is canceled by the self dual in sixdimensions ∗ w . See Appendix E for full details. • The black hole in the seed solution will likely have negative specific heat but its dilatonis stabilized which is a crucial difference with solutions considered previously. We willdiscuss what is the behavior of the solution after the rotation.In the following section we will explicitly describe the asymptotic behavior of the solution aswell as its numerical presentation in the background given by eq.(3.30).12 .2 Asymptotics
We will proceed to study the asymptotics of the equations of motion and solve them numeri-cally to obtain the new non-extremal seed solution (3.30).
A large ρ expansion of the functions ( k, q, g, h, φ ), that solve the equations of motion up toterms decaying faster than e − ρ is h ( ρ ) = e − x ( ρ ) ∼ C e − ρ ,e q ( ρ ) ∼ N c ρ −
1) + c e ρ , e g ( ρ ) ∼ N c − ρ ) + c e ρ ,e k ( ρ ) ∼ c e ρ − N c c (25 − ρ + 16 ρ ) e − ρ , e φ − φ ∼ N c c (1 − ρ ) e − ρ . (3.37)There are two parameters in the expansion above. The parameter C characterizes the non-extremal behavior. The parameter c plays a quite important role. It is one of the integrationconstants of the BPS eqs. As an expansion in inverse powers of this constant it is possibleto write a solution for the background in eq.(3.28)-see for example [10], [22]-such that whenpassed through this solution generating technique, it results in the Baryonic Branch dualsolution of [7]. Physically (and in the background before the rotation), the constant c isdescribing the coupling of the dynamics on the five-brane to gravity and ultimately to the fullstring theory, it takes the configuration out of its ‘near brane’ limit and can be thought of asindicating the insertion of an irrelevant operator in the QFT lagrangian. Near the horizon, ρ = ρ h , we expand the equations of motion in power series up to order 7.We have found, e − x ( ρ ) = x ( ρ − ρ h ) + x ( ρ − ρ h ) + x ( ρ − ρ h ) + · · · + x ( ρ − ρ h ) e q ( ρ ) = q + q ( ρ − ρ h ) + q ( ρ − ρ h ) + · · · + q ( ρ − ρ h ) e g ( ρ ) = g + h ( ρ − ρ h ) + g ( ρ − ρ h ) + · · · + g ( ρ − ρ h ) (3.38) e k ( ρ ) = k + k ( ρ − ρ h ) + k ( ρ − ρ h ) + · · · + k ( ρ − ρ h ) e ρ ) = f + f ( ρ − ρ h ) + f ( ρ − ρ h ) + · · · + f ( ρ − ρ h ) (3.39)Demanding that the expansions in eq.(3.38) satisfy the equations of motion of the systemwe determine x , ..., h ..., g ..., k ..., f ... in terms of x , h , g , k , f and thus, there are only 5 Note that the UV expansion for h ( ρ ) = e − x could have started with a lower order term such as e − ρ , seeSection 4.1 for a discussion on the choice of parameters. h 3.3 3.4 3.5 3.6 3.7 3.8 3.9 401020304050 r Figure 1:
Solution before the rotation for c = 1 . C = − e q , e k , e g and h − respectively. rh 2.0 2.5 3.0 3.5 4.0020406080100120140 r Figure 2:
Solution before the rotation for c = 125 . C = −
89. The blue, dashed,green and orange lines represent e q , e k , e g and h − respectively. independent parameters coming from the expansion at the horizon. Furthermore, x is relatedto the non-extremality parameter α -see Appendix F for the definitions, α = − x √ f g h The strategy we will follow is to numerically integrate back from infinity the equations ofmotion in Appendix F, using as boundary conditions the expansion in the UV-eq.(3.37)-andrequire that at the horizon this numerical solution matches the expansion (3.38) and its firstderivatives -up to seventh order. Doing this for a given set of values ( c, C ), determines thefree parameters ( x , h , g , k , f ). Note that not for every pair of ( c, C ) there is a blackhole solution. For some values of ( c, C ) there will not be a horizon and for others theremight be naked singularities outside the horizon. The study of the this two-parameter familyof solutions for the full ( c, C ) parameter space is an interesting question that we will notaddress here. We concentrate on finding particular values of ( c, C ) that will produce blackhole solutions. Needless to say, finding a solution before the rotation guarantees that we willhave a solution after the rotation. Examples of numerical solutions are shown in Figure 1 and2 and the respective dilatons in Figures 3 and 4. As mentioned above we integrate back from infinity and match with the expansion at thehorizon. In order to do so we define a “mismatch” function evaluated at a point ρ close to14 h 3.5 3.75 4.00.9960.9970.9980.9991.000 r (cid:227) (cid:70) Figure 3:
Dilaton, e for c = 1 . , C = − . rh 2.0 2.5 3.0 3.5 4.00.999950.999960.999970.999980.999991.00000 r (cid:227) (cid:70) Figure 4:
Dilaton, e for c = 125 . , C = − the horizon, ρ = ρ h + 0 . m ( ρ ) =[ g sh ( ρ ) − g num ( ρ )] + [ g sh ( ρ ) − g num ( ρ )] + [ h sh ( ρ ) − h num ( ρ )] +[ h sh ( ρ ) − h num ( ρ )] + [ k sh ( ρ ) − k num ( ρ )] + [ k sh ( ρ ) − k num ( ρ )] +[Φ sh ( ρ ) − Φ num ( ρ )] + [Φ sh ( ρ ) − Φ num ( ρ )] . (3.41)The subscript num denotes the numerical solution obtained by integrating back from infinityand sh denotes the series expansion (3.38) -evaluated at ρ . Recall that the expansion atthe horizon depends on five free parameters, ( x , h , g , k , f ). For a given set of ( c, C ) weminimize the mismatch function to determine ( x , h , g , k , f ). The parameters at infinity( c, C ) are adjusted so that m ( ρ ) < × − . All the numerical procedures where done inMathematica using WorkingPrecission 40. We also checked that the constraint T + U = 0(see appendix F) remains numerically negligible throughout the domain. Once we have obtained a numerical solution for the non-extremal background (3.30) we caneasily generate the rotated solution as summarized in eq.(3.35) and outlined in the AppendixE. Before presenting the numerical results let us look at the large ρ asymptotics. Using the UVexpansions of the seed solution (3.37), one can obtain the black hole metric (3.35) asymptotics15 fter the rotation, − g tt ∼ e / ρ A ( ρ ) − e − / ρ c √ A ( ρ ) [256 c C + 27(1 − ρ ) + 96 c C ( − ρ )] + · · · ,g xx ∼ e / ρ A ( ρ ) − e − / ρ c √ A ( ρ ) [27(1 − ρ ) ] + · · · ,g ρρ ∼ c A ( ρ ) + · · · , g θθ ∼ c A ( ρ ) + e − / ρ A ( ρ )(2 ρ −
1) + · · · ,g ψψ ∼ c A ( ρ ) + · · · , g ˜ θ ˜ θ ∼ c A ( ρ ) − e − / ρ A ( ρ )(2 ρ −
1) + · · · . (3.42)Where we have defined the quantity c √ A ( ρ ) ≡ p ρ − C c − . (3.43)Using a variable u = e ρ/ the metric can be written -to leading order- as, ds = u A ( u ) ( dx i dx i ) + 3 cA ( u )2 ( du u + ds T ) + · · · O ( u − ) . (3.44)Thus, the non-extremal solution found after the rotation has the asymptotics of the Klebanov-Strassler background [8]. We point out that the reason we get precisely Klebanov-Tseytlintype of asymptotics lies in our precise arrangements of coefficients in the seed solution. Namelywe arranged for a seed solution with no linear dilaton and with a dilaton behavior that, inthe field theory limit, eliminates the leading term in the harmonic function given in equation(3.36).With our careful choice of asymptotic conditions for the seed solution of D5 branes wehave managed to construct black holes in asymptotically KT backgrounds. Our solution isdifferent from those obtained in the traditional approach to black holes in asymptotically KTbackgrounds (see, for example, [33]). One glaring difference is the form of the RR fluxes. Inparticular the presence of the non-extremality parameter in F and in the warp factor H ( ρ )is completely novel.In Figures 5-8 we present the numerical plots of the diffent metric elements after the rotationfor two sets of parameters. In Figures 9 and 10 we see how the angular part of the solutionapproaches the T , metric for the two representative solutions. Let us clarify the expectations of the form of the background that arise from field theoryconsiderations. In the previous section we obtained a solution largely characterized by twoparameters ( c, C ). After the rotation, the interpretation of c has been spelled out explicitly16 h 2 2.5 3 3.5 4.0100015002000250030003500 Figure 5:
The g rr metric element for a so-lution after the rotation, c = 125 . , C − rh 3.4 3.6 3.8 4.00500100015002000 Figure 6:
The g rr metric element for a solu-tion after the rotation, c = 1 . , C = − rh 2 2.5 3 3.5 4.005101520 Figure 7: g xx and g tt for a solution after therotation, c = 125 . , C = − rh 3.4 3.6 3.8 4.00.00.51.01.52.02.53.0 Figure 8: g xx and g tt for a solution after therotation, c = 1 . , C = − Figure 9:
Angular part of the metric afterthe rotation. We plot g θθ , g ˜ θ ˜ θ and g ψψ . So-lution with c = 125 , C = − Figure 10:
Angular part of the metric afterthe rotation. We plot g θθ , g ˜ θ ˜ θ and g ψψ . So-lution with c = 1 . , C = −
17n [15], [6] and [13]. It is related to the vev of the baryon operator. The other parameterpresent in the solution is C , its interpretation coincides roughly with the temperature of thesolution in the field theory.We will present and exhaustive analysis of these classes of solutions elsewhere and will limitour attention here to a few main points. The main deterrent in presenting a final descriptionof the thermodynamics is the need to explore a large parameter space of the solutions as wellas the difficulty in matching data at asymptotic infinity with data near the horizon. Theanalysis is numerically costly but we emphasize that these are not difficulties of principle.Our goal in this work is to explicitly show, through numerical analysis, the existence of theconjectured solutions.Let us show that the thermodynamic analysis can, in principle, be done explicitly for thesolutions we have. For the temperature, we use the standard approach of looking at the Eu-clidean section and imposing the absence of conical singularities to determine the temperatureof the supergravity backgrounds. Basically, for a metric with Euclidean section of the form ds = f dτ + dρ g + . . . , (3.45)we find a temperature equal to T = 14 π p f g , (3.46)where prime represents derivative with respect to the radial coordinate and the above expres-sion is evaluated at the horizon. Interestingly, the temperature in both backgrounds, that is,before and after the rotation, is the same. In terms of the near horizon data we have T = 14 π x √ k . (3.47)It is worth noting that this is the temperature at the horizon. In the case of asymptoticallyflat backgrounds (before the rotation) we have the option of computing the temperature asseen by an observer at infinity. We refer the reader to [23] for a detailed discussion of thisprocedure in the D5 or NS5 background. The entropy is computed, as usual, as the area ofthe horizon. The most complicated quantity to compute is the free energy which is requiredto compute, for example, the specific heat. For the free energy we need to evaluate the action.In the case of asymptotically flat solutions for D5/NS5 branes we can essentially follow [23].A very preliminary analysis seems to confirm our intuition about the specific heat of suchsolutions. Namely, evaluation of the action suggests that before the rotation the solutionsobtained have negative specific heat. A precise evaluation of the action after the rotationis more subtle as naive evaluation yields divergent terms. It is likely that a more accurateapproach requires some sort of holographic renormalization or substraction along the lines of[33]. Preliminary evaluation suggests a positive specific heat for the rotated solutions . Dario Martelli suggested that along the lines of the first two papers in [3], some thermodynamical quantitiesshould be invariant under T-dualities, shifts of coordinates and boosts. See also [34]. About decoupling limits
It is worth discussing a bit about decoupling limits. Let us do this in the extremal case, theextension to the cases with non extremal factors follows from what we write here. The firstpoint to emphasize is that when gravity modes are coupled to a generic D-brane configuration,the warp factors asymptote to a constant. An example of this is what happens to the flat Dpbranes, with H = 1 + ( Qρ ) − p .We will compare warp factors in the examples studied in the previous sections. It is enoughto focus on the expressions of the metric component g tt . In all the examples studied we found g tt = 4 AB − C √ B . (4.48)Indeed, in the case in which we rotate a flat Dp-brane, we have eq.(A.6) with
A, B, C given ineq.(A.5). In the case of the rotation in p − q directions the results for A, B, C are written ineq.(2.25) and in the case of the wrapped D5’s are given in eq.(E.9). Again, for the point wewish to emphasize, we will focus on the extremal cases, when the rotation preserves SO (1 , p ).So, in these cases we have , g tt, ( p − q ) = − β p H − tanh β , g tt,p = − √ H q H ( ˆ A tanh β − − tanh β ,g tt,D = − β p e − φ − tanh β (4.49)The case of the rotation in p directions shows that the induced charge of D0 brane (after thefirst set of T-dualities) as shown by eq.(A.2), enters in the new warp factor via the one-formpotential A = ˆ Adt . This makes the decoupling of the final configuration more unclear andtypically, if we start with ‘decoupled’ Dp branes (that is H ∼ ρ p − ) we will generate a non-decoupled configuration, with an asymptotically constant warp factor as discussed aroundeq.(2.13). Of course, we can try to take the decoupling limit in the final configuration.On the other hand the cases of the wrapped D5’s and the rotation in p − q directions(that share the fact that a D0 charge is induced by the boost only) are such that we must start with a non-decoupled configuration in order to avoid a generated background with asingularity or end of the space in the UV (zero warp factor at some finite value of the radialcoordinate). Indeed, if H = 1 + ( Qρ ) − p we choose a particular value of the boost parameter β → ∞ , together with a particular choice of the value of e φ ( ∞ ) = 1 in the D5’s case) to obtaina background decoupled from gravity . Gregory Giecold correctly pointed out that in the non-extremal cases the non-extremality factor h ( ρ )enters as expected, for example g tt,D = − h cosh β √ e − φ − h tanh β . We thank him for this and various othercorrect comments. One of the interesting observations in [6], is that in the case of the D5’s the solution described in the .1 Choice of Parameters for N = 1 Black Holes
Let us discuss in more detail the choice of parameters in the solution. This is a particularlysensitive question from the numerical and from the physical points of view. We note that apriori there are ten parameters in our solution as corresponds to a system of five second orderordinary differential equations. The constraint lowers this number by one. At this point thesimplest route for us is to notice that we are looking for a particular solution that asymptotesto the extremal supersymmetric one found in [10],[13] (see, in particular, appendix B there).In that context most of our constants get fixed, as we require the same UV behavior; we usethe expressions in that solution to fix six constants. Of the three constants left, one gets fixedby the asymptotic behavior of the non-extremality function, that is, we choose h ( ρ ) ∼ − C e − ρ . (4.50)The leading coefficient guarantees that we will have, after rotation, a solution with KT asymp-totics. The key observation to be made is that, in principle, there is a mode allowed that wehave explicitly set to zero. Namely, another solution to the equations of motion in the asymp-totic regime could be h ( ρ ) ∼ C e − ρ − C e − ρ + . . . . (4.51)The above expression should be thought of schematically as the solution to a second orderdifferential equation for h ( ρ ) which is defined asymptotically by two parameters. This C is one of the three constants that we fix and therefore we remain with two constants thatwe called ( c, C ). The parameter c has the same meaning discussed in [10],[13] and C isrelated to the temperature. Thus, the condition that the solution after the rotation has KTasymptotics plus a condition on the non-extremality function determine the values of theintegration constants.It is worth noting that selecting the right constants in the D3/D5 picture is more cumber-some as there are more parameters in the generic D3/D5 approach to black holes with KTasymptotics. For example, our solution does not allow for a generic mode that will lead toa Freund-Rubin form for F . As has been repeatedly emphasized, our F is related to thenon-extremality function and deforms the typical term proportional to the volume 5-form.Let us explicitly visualize the difficulty in choosing the asymptotic expansion from the D3/D5point of view in another example. The choice C = 0 roughly corresponds to a dimension3 operator in the D3/D5 picture. This can be seen easily by going to the radial coordinate u = exp(2 ρ/ papers [10], [22] is such that the dilaton asymptotes to a (tunable) constant and the subleading terms areprecisely those needed to generate the warp factor of Klebanov-Strassler Conclusions and Future Directions
We present here some concluding remarks together with some possible future projects tocomplement and refine the material in this paper.In this work we have elaborated on the U-duality proposed in [6]. In that case, it was usedto connect solutions describing wrapped D5 branes with the family of solutions describing thestrong coupling dynamics of the baryonic branch of the Klebanov-Strassler field theory. In ourcase, we have used this U-duality to construct a new family of solutions describing the finitetemperature phase of this baryonic branch. But before doing so, we explored the U-duality(that, following custom, we called rotation) in more generality, as a solution generating tech-nique. We applied it to flat Dp branes in two different circumstances so that we appreciatedthe subtle differences making the choice of dualities in [6] so special.Once we developed this technology, we applied it to rotate the Witten model for a Yang-Millsdual [17]. All this represents new material and it would be interesting to find an applicationfor it, hopefully in the context of duals to Yang-Mills or QCD-like field theories.In section 3, we discussed various new things: first a class of non-extremal solutions describ-ing five branes wrapping a two-cycle inside the resolved conifold. The main feature of thesenon-extremal backgrounds is that the dilaton is asymptotically stabilized, which we inter-preted as coupling the dual field theory to gravity modes. Then, we applied the U-duality tothis solution, to find a new background of IIB that describes a non-extremal deformation of theKlebanov-Tseytlin solution ( decoupled from gravity or stringy modes). We briefly commentedon the associated thermodynamic properties of this new family of backgrounds. Finally, wediscussed various aspects of this coupling/decoupling from gravity modes in section 4.Let us now list a set of projects that complement the material presented here and that mayimprove the understanding of the topic: • It is important to sort out the details of the thermodynamics after the rotation in thesolutions presented in section 3. Indeed, as we indicated in section 3.5, this poses anumerical problem, but certainly does not amount to a conceptual obstruction. Theresult for quantities like the specific heat or the free energy will be of great interest.Notice that this presents a novel way to approach the finite temperature phase of theKlebanov-Strassler field theory. Comparison with the solutions of [38] would be inter-esting. • More conceptually; it would be nice to better understand the role played by the D0branes in this solution generating technique. Indeed, notice the differences between theexamples analyzed in section 2 and the one in section 2.1, all due to the presence ofD0 branes before the boost that translates into the different final results. In the samevein, it would be good to have a better understanding on the reasons for the equivalencebetween the U-duality explained here and the rotation of the J , Ω forms as described21n [12], [13]. • It would be of very much interest to extend our non-extremal solution to a background ofthe form in eq.(3.28) where the restriction a ( ρ ) = b ( ρ ) = 0 is not applied. Indeed, findingsuch a non-extremal solution would, after the application of the U-duality, produce asolution dual to a non-SUSY KS-like field theory in the baryonic branch. It may proveinteresting to compare this new solution with that presented in the papers [35]. Thedifferent nature of the non-SUSY deformation will certainly reflect in different fieldtheory aspects. • It would be nice to use the solution presented in section 3 as a base to produce anon-SUSY deformation along the line proposed recently in [36]. The authors of [36]are finding SUSY breaking backgrounds by adding anti-D3 branes. Their solutionshave singularities (and the problem is if such a singularity is physically acceptable).Proceeding as with the solution in this paper may set us on a different branch of solutions. • It would be interesting to find a Physics application of the new backgrounds of section2.1.1. • It would be nice to extend the treatment here to the case in which the seed backgroundcontains N f ∼ N c flavors. This would be following the lead of the papers [24], [25]. Thesubtle point would be how to rotate such solutions (in other words, how the sourcesget represented in M-theory, see [37]. Comparison with the solutions of [38] would beinteresting) • We have presented two numerical solutions as examples of the type of backgroundsobtained. These examples correspond to specific values of the UV parameteers. We didnot attempt a study of the full family of solutions in the whole parameter space. Thisis an interesting question worth investigating.We hope to address some of these problems in the near future.
Acknowledgments:
We would like to thank various colleagues for the input that allowed us to better understandand present the results of this paper. We would like to single out Eloy Ay´on-Beato, Iosif Bena,Alex Buchel, Pau Figueras, Gregory Giecold, Mariana Gra˜na, Prem Kumar, Dario Martelli,Ioannis Papadimitriou, Michela Petrini, Dori Reichmann and C´esar Terrero-Escalante. Veryspecial thanks to Gregory Giecold for a very careful reading of the manuscript. E.C. thanksthe Theory Group at the University of Texas at Austin for hospitality.22his work was supported in part by the National Science Foundation under Grant Num-bers PHY-0969020 and PHY-0455649, the US Department of Energy under grant DE-FG02-95ER40899 and CONACyT grant CB-2008-01-104649.
A Appendix: U-duality in the case of flat Dp-branes
In this appendix we will give a sketchy derivation of the U-duality chain, in the case in whichwe rotate flat Dp-branes. We will not consider the non-extremal case, this can be done easilyand the result of doing so is included in the first section of the paper. Here we want just toemphasize some points.We start from the SUSY configuration in type II supergravity (string frame), ds = H − / ( − dt + dx p ) + H / ( dρ + ρ d Ω − p ) ,e φ = H − p , F p +2 = ∂ ρ ˆ Adt ∧ dx ∧ ...dx p ∧ dρ (A.1)where ˆ A = − A in the notation of Section 2. Following standard convention we define thecharge of F − p as positive. The sign of F p +2 will then depend on whether we considerMinkowski or Euclidean worldvolume on the D p brane. We apply T-duality in the x p di-rections, ds = − H − / dt + H / ( dx p + dρ + ρ d Ω − p ) ,e φ = H , F = ∂ ρ ˆ Adt ∧ dρ (A.2)We lift this to M-theory, using e φ = H , ds = H ( dx + ˆ Adt ) − H − dt + dM , dM = ( dx p + dρ + ρ d Ω − p ) (A.3)we now boost in the t − x direction with rapidity βdt → cosh βdt − sinh βdx , dx → − sinh βdt + cosh βdx , (A.4)to get ds = Adt + Bdx + Cdtdx + dM ,A = H − [ H ( ˆ A cosh β − sinh β ) − cosh β ] B = H − [ H ( ˆ A sinh β − cosh β ) − sinh β ] C = − H − [ H ( ˆ A cosh β − sinh β )( ˆ A sinh β − cosh β ) − cosh β sinh β ] (A.5)and we prepare the metric for reduction back to IIA as ds = B ( dx − a t dt ) + B − / h g tt dt + B / dM i ,a t = − C B , g tt = 4 AB − C √ B (A.6)23e now write the IIA configuration as, ds IIA = g tt dt + B / ( dx p + dρ + ρ d Ω − p ) , e φ = B / , F = ∂ ρ a t dt ∧ dρ (A.7)and we T-dualize back in the x p directions to get ds II = g tt dt + B − / dx p + B / ( dρ + ρ d Ω − p ) ,e φ = B − p , F p +2 = ∂ ρ a t dt ∧ dρ ∧ dx .... ∧ dx p (A.8)We see that the SO (1 , p ) invariance is preserved, as g tt √ B = − B Appendix: Black Dp branes
In this appendix we gather some useful facts about black Dp-brane solutions [26],[27],[28].The metric in string frame is, ds = − f + ( r ) p f − ( r ) dt + p f − ( r ) p X i =1 dx i dx i + f − ( r ) − − − p − p f + ( r ) dr + r f − ( r ) − − p − p d Ω − p (B.1)where f ± ( r ) = 1 − (cid:16) r ± r (cid:17) − p (B.2)The mass per unit volume M and the R-R charge N are M = 1(7 − p )(2 π ) d p l P ((8 − p ) r − p + − r − p − ) , N = c p ( r + r − ) − p , (B.3)where l P is the ten dimensional Planck length l P = g / s l s and c p , d p are numerical factors, d p = 2 − p π − p Γ (cid:18) − p (cid:19) c p = 1 d p g s l − ps = (7 − p ) V ol S − p (2 πl s ) − p g s (B.4)The background also has a nontrivial dilaton and a R-R flux, e − = g − s f − ( r ) − p − πl s ) − p Z S − p ∗ F p +2 = N. (B.5)The Einstein frame metric ( g E µν = p g s e − Φ g st µν ) has a horizon at r = r + and, for p ≤
6, asingularity at r = r − . If r + > r − the singularity is covered by the horizon and the solution isa black hole. In the extremal case, r + = r − , the space is singular except for p = 3.24et us define a new coordinate, ρ − p = r − p − r − p − . This change of variables transforms (B.1) to a more familiar form, ds = H ( ρ ) − − h ( ρ ) dt + p X i =1 dx i dx i ! + H ( ρ ) (cid:18) dρ h ( ρ ) + ρ d Ω − p (cid:19) (B.6)where H ( ρ ) = 1 + r − p − ρ − p h ( ρ ) = 1 − r − p + − r − p − ρ − p (B.7)Throughout the paper we use the notation, L − pp ≡ r − p − and R − pT ≡ r − p + − r − p − . Note that using (B.3) one can also write L − pp = ˜ Q s (cid:18) R − pT Q (cid:19) − R − pT Q = ˜ Q α p . (B.8)where ˜ Q = Nc p and α p = (1 + ( R − pT Q ) ) / − R − pT Q .¿From (B.3) we can see that the parameter R T is related to the energy per unit volumeabove extremality (cid:15) , ∆ M = (cid:15) = r − p + − r − p − (2 π ) d p l P = R T − p (2 π ) d p l P . (B.9)Note that in the decoupling limit ( α →
0, energies fixed) (cid:15) remains fixed and α p → r + = r − , we have R T = 0 and L − pp = ˜ Q = N d p g s l − ps . C Appendix: Some details on the U-duality
Here we provide details for the derivations of eq.(2.24). We started with ds = H − / [ − dt + dx p − q + dσ q ] + H / [ dρ + ρ d Ω − p ] ,F p +2 = ∂ ρ ˆ Adt ∧ dx ∧ ... ∧ dx p − q ∧ dσ q ∧ dρ,e φ [ initial ] = e φ ( ∞ ) H − p . (C.1)25he we perform the U-duality described in the text below eq.(2.23) ds = H − / [ − dt + dσ q ] + H / [ dρ + ρ d Ω − p + dx p − q ] ,F q +2 = ∂ r ˆ Adt ∧ dσ q ∧ dρ,e φ = e φ ( ∞ ) H − q . (C.2)Now, we need this to be a configuration in Type IIA (in order to lift this to M-theory). Wewill also want to impose that when lifted to M-theory this will produce a four-form field G .If this is the case, we must have that either q = 2 or that q = 4 (the cases of q = 0 , q = 6are analogous to what we analyzed in the first section). The case q = 2, on which we willelaborate upon below, has the peculiarity that the reduction from eleven dimensions back toIIA will generate a NS H field, proportional to the boost. Let us see this in detail. To beginwith, we will lift to eleven dimensions the configuration in eq.(C.2). ds = H / dx + H − / [ − dt + d~σ ] + H / [ dρ + ρ d Ω − p + d~x p − ] ,G = ∂ ρ ˆ Adt ∧ dρ ∧ dσ ∧ dσ (C.3)and now we boost with rapidity βdt → cosh βdt − sinh βdx , dx → − sinh βdt + cosh βdx (C.4)and we can rewrite the configuration after the boost as, ds = Adt + Bdx + Cdtdx + dM ,G = ∂ ρ ˆ A (cosh βdt − sinh βdx ) ∧ dρ ∧ dσ ∧ dσ ,dM = H − / d~σ + H / [ dρ + ρ d Ω − p + d~x p − ] ,A = H − / [ H sinh β − cosh β ] ,B = H − / [ H cosh β − sinh β ] ,C = 2 H − / sinh β cosh β (1 − H ) (C.5)when we reduce this to IIA we get, ds IIA,st = g tt dt + B / (cid:16) H − / d~σ + H / [ dρ + ρ d Ω − p + d~x p − ] (cid:17) ,e φ [ A ] = B / , g tt = 4 AB − C √ B ,F = ∂ ρ ˆ A (cosh β + a t sinh β ) dt ∧ dρ ∧ dσ ∧ dσ ,H = sinh β∂ ρ ˆ Adρ ∧ dσ ∧ dσ ,F = ∂ ρ ( a t ) dt ∧ dρ, a t = C B = sinh β cosh β (1 − H ) H cosh β − sinh β , (C.6)26e see that we have generated a NS magnetic field. Finally, we T-dualize back in the ~x p − directions, to get ds II,st = g tt dt + d~x p − H / B / + B / (cid:16) H − / d~σ + H / [ dρ + ρ d Ω − p ] (cid:17) ,F p +2 = ∂ ρ ˆ A (cosh β + a t sinh β ) ∧ dρ ∧ dσ ∧ dσ ∧ dx ∧ .... ∧ dx p − ,H = sinh β∂ ρ ˆ Adρ ∧ dσ ∧ dσ ,F p = ∂ ρ ( a t ) dt ∧ dρ ∧ dx ∧ .... ∧ dx p − ,e φ [ final ] = B − p H − p . (C.7)This completes our derivation of eq.(2.24). D Appendix: Another solution generating algorithm
We studied two different ‘solution generating techniques’ and applied them to different back-grounds. All these ‘algorithms’ were starting with a background solution to the Type II (Aor B) equations of motion, applying a number of T-dualities that would transform the back-ground into a solution for Type IIA supergravity. Then we lifted this to eleven dimensions,where a boost was applied (inducing a one parameter- β - family of solutions), then reducingto IIA and T-dualizing back we had our final generated background.One may wonder what is the algorithm when, after a number of T-dualities, we end witha background solving the Type IIB supergravity equations of motion. In this case, a way ofgenerating a one-parameter family of solutions may be S-dualizing. Indeed, given a, b, c, d realnumbers satisfying ad − bc = 1, we can start with a IIB solution having axion χ , dilaton φ , RRand NS thre forms F = dC and H = dB and five form F = dC + F ∧ B and by S-duality,generate a new solutions dependent on the three independent real parameters a, b, c . Indeed,the five form is left invariant and the same happens for the Einstein frame metric, while F [ new ] = bH + aF , H [ new ] = dH + cF ,e φ [ new ] = (cid:16) ( cχ + d ) + c e − φ (cid:17) e φ , χ [ new ] = ( aχ + b )( cχ + d ) + ace − φ (cid:16) ( cχ + d ) + c e − φ (cid:17) (D.1)Let us focus our attention in the particular set of values a = d = 0 , cb = −
1, this is thetransformation that interchanges the three forms and inverts the value of the dilaton (if theaxion is initially zero, as we will assume).One can then think about applying this transformation expecting to generate new inter-esting solutions. As an example, suppose that we start with solutions describing D6 braneswrapping a three-cycle inside the deformed conifold. Those solutions are dual to a (UV com-pleted version of) N=1 SYM [29]. A particularly interesting solution is given in section 3.2 of27he paper [30]. The background consist of metric, dilaton and RR two-form field, ds IIA,st = e f [ − dt + dx + dx + dx + dM ] , e φ = e f +2 φ , F = dA (D.2)where all the details (the manifold M , the functional form of f, φ and the one form A aregiven in eqs(57)-(59) of [30]). We can the perform three T-dualities in x , , leading to ds IIB,st = e f [ − dt + dM ] + e − f dx , , , e φ = e φ ,F = dA ∧ dx ∧ dx ∧ dx (D.3)that is we have generated D3 brane charge and the dilaton is just a constant. We now movethis to Einstein frame, that leaves us-up to a constant-with the same metric, perform the S-duality metioned above and T-dualize back. This brings us to the starting point background(D.2) . Something similar occurs if we start with a solution describing D5 branes wrapping athree-cycle inside a G2 holonomy manifold [31], a configuration dual to N=1 Yang-Mills Chern-Simons in 2+1 dimensions (with its respective UV completion). Notice that this operation isnot the one proposed in [20], that is the reason why these authors were able to generate aninteresting solution starting from [31].In both these cases described above, we are generating charge of D3 brane, this is invariantunder the S-duality, hence it is expected that the whole operation brings us back to the initialconfiguration. A more interesting example is to start from the configuration of D4 braneswrapping a circle with SUSY breaking boundary conditions [17] that we discussed before, seeSection 2.1.1.In this case we will apply three T-dualities to a IIA configuration. This will generate chargeof D1 brane that the S-duality will interchange with F1 charge, after T-dualizing back, we willgenerate a new background (the SO (1 ,
3) isometry will be spoiled, so we may want to startwith the high Temperature dual, hence having a non-extremal factor h ( ρ ) in front of dt andthe function f ( ρ ) = 1 in front of dx , but let us keep things general). Let us see some details.After the T-dualities, the configuration reads ds IIB,st = H − / ( − hdt + f dx ) + H / ( dρ h + ρ d Ω + dx , , ) ,e φ = g s H, F = ∂ ρ Adt ∧ dx ∧ dρ (D.4)now, we need to move this to Einstein frame, multiplying the metric by e − φ , perform theS-duality that will generate H and a dilaton e − φ = g s H . Then, T-dualize back in x , , . Thefinal configurations is, ds IIA,st = c / g s H ( − hdt + f dx ) + dρ h + ρ d Ω + dx , , ,H = c∂ ρ Adt ∧ dx ∧ dρ, c e − φ = g s H. (D.5) One may perform an S-duality with parameters a = 0 , bc = − d being free. This will generate abackground in IIB with constant axion-dilaton. The final IIA configuration is the same as the initial one.
28s a final remark; had we chosen to T-dualize only in the x direction (in the supergravityapproximation, this can be done in the high Temperature phase only), we would have gen-erated D3 branes, hence after the S-duality and T-duality, we would be back to the initialconfiguration. E Appendix: U-duality for the wrapped D5 branes BlackHole
We will describe in detail the action of the solution generating technique proposed in [6],when applied to the background of eq.(3.30). These techniques have been applied in a similarcontext in [39]. Let us consider things in the string frame. ds s = e φ h − h ( ρ ) dt + dx + dx + dx i + ds ,s ds = e φ h e k s ( ρ ) dρ + e k ω + cos θdϕ ) + e q ( dθ + sin θdϕ ) + e g d ˜ θ + sin ˜ θd ˜ ϕ ) i ,F (3) = N c " − ˜ ω ∧ ˜ ω + sin θdθ ∧ dϕ ∧ (˜ ω + cos θdϕ ) , (E.1)where h ( ρ ) , s ( ρ ) are the non-extremality functions and we will choose h ( ρ ) = s ( ρ ) as we didin section 3. We can proceed to rotate it. We will follow the procedure explained in [6]. So,let us start by writing the effect of the first T-duality in the x direction (all the expressionsbelow are in string frame) ds IIA = e f h − hdt + dx + dx i + e − f dx + ds ,e φ A = e φ − f , F = F ∧ dx (E.2)The function f = φ will be kept to avoid confusion with the transformed dilatons. Now, weperform the T-duality in x ds BIIA = e f h − hdt + dx i + e − f ( dx + dx ) + ds ,e φ B = e φ − f , F = F ∧ dx ∧ dx (1 + ∗ ) (E.3)T-dualizing in x , we get ds IIA = e f h − hdt i + e − f ( dx + dx + dx ) + ds ,e φ A = e φ − f ,F = F ∧ dx ∧ dx ∧ dx → F = e f √ h ∗ F ∧ dt (E.4)29otice that ∗ F = N c √ h h − e q − g sin θdθ ∧ dϕ + e g − q θd ˜ θ ∧ d ˜ ϕ i ∧ dρ. (E.5)The factor of √ h in the F of eq.(E.4) is present to cancel the factor of √ h in the denominatorof eq.(E.5). Now, we lift this to M-theory; ds = e / φ − f dx + e f − / φ h − he f dt + e − f ( dx + dx + dx ) + ds i ,G = √ he f ∗ F ∧ dt. (E.6)We boost in the t − x directions according to, dt → cosh βdt − sinh βdx , dx → − sinh βdt + cosh βdx (E.7)and now we rewrite this boosted metric as, ds = e f − / φ h e − f ( dx + dx + dx ) + ds i + Adt + Bdx + Cdtdx ,G = √ he f ∗ F h cosh βdt − sinh βdx i (E.8)where, A = e f − / φ [sinh βe φ − f − h cosh β ] , B = e f − / φ [cosh βe φ − f − h sinh β ] ,C = − β sinh βe f − / φ [ e φ − f − h ] . (E.9)Now, we will reduce this to IIA, before doing so and in order to reduce to IIA, it is useful torewrite eq.(E.8) as, ds = B − / h g tt dt + B / e f − / φ ( e − f ( dx + dx + dx ) + ds ) i + B ( dx + a t dt ) ,G = √ he f ∗ F h (cosh β + a t sinh β ) dt − sinh β ( dx + a t dt ) i (E.10)where we have defined a t = C B , g tt = 4 AB − C √ B , e / φ A = B. (E.11)Now, we reduce to IIA, obtaining in string frame, ds IIA = g tt dt + √ Be − / φ ( dx + dx + dx ) + √ Be f − / φ ds ,e φ A = B / ,F = √ he f ∗ F ∧ h (cosh β + a t sinh β ) dt i ,H = − sinh β √ he f ∗ F ,F = a t dρ ∧ dt (E.12)30ow, we proceed to do the T-dualities back; T-dualizing in the x direction we have ds IIB = g tt dt + √ Be − / φ ( dx + dx ) + e / φ √ B dx + √ Be f − / φ ds ,e φ B = Be / φ ,F = √ he f ∗ F ∧ h (cosh β + a t sinh β ) dt ∧ dx i (1 + ∗ ) ,H = − sinh β √ he f ∗ F ,F = a t dρ ∧ dt ∧ dx (E.13)now, we T-dualize in x ds IIA = g tt dt + √ Be − / φ ( dx ) + e / φ √ B ( dx + dx ) + √ Be f − / φ ds ,e φ A = √ Be / φ ,F = √ he f ∗ F ∧ h (cosh β + a t sinh β ) dt i ∧ dx ∧ dx ,H = − sinh β √ he f ∗ F ,F = a t dρ ∧ dt ∧ dx ∧ dx (E.14)finally, we T-dualize in x ds IIB = g tt dt + e / φ √ B ( dx + dx + dx ) + √ Be f − / φ ds ,e φ B = e φ ,F = √ he f ∗ F ∧ h (cosh β + a t sinh β ) dt i ∧ dx ∧ dx ∧ dx F = ∗ F ,H = − sinh β √ he f ∗ F ,F = a t dρ ∧ dt ∧ dx ∧ dx ∧ dx (1 + ∗ ) . (E.15)After using f = φ and the definitions for A, B, C, a t this encodes the result of eq.(3.35). F Appendix: The equations of motion
In this appendix we will quote the equations of motion that we are numerically solving. Ourgoal is to find a black hole of the metrics described in the main text. In [23] the authorsstudied a very general Ansatz for non-extremal deformations of
N S S .Their Ansatz can be adapted to our case. It reads –in Einstein frame, ds = − Y dt + Y d x n d x n + Y dρ + Y ( e + e ) + Y (cid:0) ( w ) + ( w ) (cid:1) + Y ( w + A ) (F.1)31nserting this ansatz in the supergravity action we get, L = X i,j G ij ( Y ) Y i Y j − U ( Y ) = T − U (F.2)Using a parametrization to make G ij diagonal and choosing the appropiate gauge to makecontact with our ansatz we have, T = e g + q − x +Φ) (cid:18) (cid:0) g + q − (cid:1) + 2( g q − k x ) + ( g + q + Φ )( k − x + 2Φ ) (cid:19) ,U = 1256 e − g + q − Φ) (cid:16) − e g + q + k ) ( e g + 4 e q ) + ( e g + 16 e q )(1 + e k ) (cid:17) . (F.3)and, Y = e Φ / e − x , Y = e Φ / , Y = e Φ / e x e k ,Y = e Φ / e g , Y = e Φ / e q . Y = Φ (F.4)¿From (F.2) we get the second order equations to solve,2 e − g +8 x + 18 e − q +8 x − ( g + q − x + Φ ) − Φ = 0 − x ( g + q − x + Φ ) − x = 0 − e k +8 x (2 e − g + 18 e − q ) + e x (2 e − g + 18 e − q ) + 2 k ( g + q − x + Φ ) + k = 0 (F.5)2 e x ( e k − g − e k − g ) + 1) + g ( g + q − x + Φ ) + g = 0 e x ( − e k − q + e k − q − e − q ) − q ( g + q − x + Φ ) − q = 0and a first order constraint which is a consequence of reparametrization invariance, e x (cid:16)
116 ( e − q + 16 e − g )(1 + e k ) − e k ( e − q + 4 e − g ) (cid:17) + 12 ( g + q − )+2( g q − k x ) + ( g + q + Φ ) ( k − x + 2Φ ) = 0 (F.6)The second equation in (F.5) can be integrated to yield a first order equation and (F.5)becomes 2 e − g +8 x + 18 e − q +8 x − ( g + q − x + Φ ) − Φ = 0 x − αe − g + q − x +Φ) = 0 − e k +8 x (2 e − g + 18 e − q ) + e x (2 e − g + 18 e − q ) + 2 k ( g + q − x + Φ ) + k = 0 (F.7) e x ( e k − g − e k − g ) + 1) + g ( g + q − x + Φ ) + g = 0 e x ( − e k − q + e k − q − e − q ) − q ( g + q − x + Φ ) − q = 032here α is a non-extremality parameter.One can verify that the transformation r → e d r Φ → Φ + d x → x + d g → g q → q k → k (F.8)where d is a constant, leaves (F.5) and ( F.6) invariant and thus, is a symmetry of the equationsof motion and constraint. Another, obvious, symmetry of (F.5) and ( F.6) isΦ → Φ +
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Elena C´aceres ∗ , Carlos N´u˜nez † and Leopoldo A. Pando Zayas ∗∗ ∗ Facultad de CienciasUniversidad de ColimaBernal Diaz del Castillo 340, Colima, M´exico. andTheory Group, Department of Physics,University of Texas at Austin, Austin, TX 78727, USA. † Department of PhysicsUniversity of Swansea, Singleton ParkSwansea SA2 8PPUnited Kingdom. ∗∗ Michigan Center for Theoretical PhysicsRandall Laboratory of Physics, the University of MichiganAnn Arbor, MI 48109-1040. USA
Abstract
We study different aspects of a U-duality recently presented by Maldacena and Martelliand apply it to non-extremal backgrounds. In particular, starting from new non-extremalwrapped D5 branes we generate new non-extremal generalizations of the Baryonic Branch ofthe Klebanov-Strassler solution. We also elaborate on different conceptual aspects of theseU-dualities, like its action on (extremal and non-extremal) Dp branes, dual models for Yang-Mills-like theories, generic asymptotics and decoupling limit of the generated solutions. [email protected] [email protected] [email protected] a r X i v : . [ h e p - t h ] F e b ontents N = 1 SUSY Solutions 9 N = 1 Black Holes . . . . . . . . . . . . . . . . . . . 20 Introduction
The Maldacena conjecture [1] and some of its refinements [2] are the guiding principle be-hind much of the progress in the interface String Theory-Quantum Field theory in the lasttwelve years. The influence of this approach extends to toy models with different numberof supersymmetries (SUSY’s), systems at finite temperature and/or finite density and lowerdimensional systems. The applications to physically relevant systems do not seem to beexhausted and on the contrary, increase with time. In this sense, finding new (trustable)solutions to the equations of motion of String Theory (even in the point particle and classicalapproximation) has become an industry with various applications in Physics and Mathematics.Solution generating techniques have certainly played a role. Some examples worth mentioningare the combination of T-dualities and shifts of coordinates that generated solutions dual tominimally SUSY superconformal field theories (beta deformations) or duals to non-relativisticfield theories [3] .In this paper we will focus our attention on a particular solution generating technique thatwas presented in [6]. The procedure suggested by these authors starts by taking a solution intype IIB string theory proposed to be dual to (a suitably UV-completed version of) N = 1Super Yang-Mills in four dimensions. The solution has the topology R , × M where M preserves four supercharges. The algorithm to generate the new solutions goes as follows: firstapply a set of three T-dualities in the R directions, which leaves us with a IIA configuration;lift this configuration to M-theory and boost–with rapidity β –in the eleventh direction; thenreduce to IIA and T-dualize back in R . This generates a solution that roughly speaking isthe dual description to the baryonic branch [7] of the Klebanov-Strassler field theory [8]. Forall the technical details of this procedure the reader can refer to [6] or to our Appendix E.The ‘seed solution’ (as we will refer to the initial solution on which the generating algorithmis applied) was discussed in the papers [9] (Section 8) and [10] (Section 4.3). In a particularlimit, this becomes the exact solution discussed in [11].We will also refer to the solution generating algorithm, that is a U-duality, as ‘rotation’in a sense explained below and in previous papers. In the following we will emphasize someaspects of the generating technique presented in [6] that we find particularly interesting andhave not been explicitly discussed in the existing bibliography: • In the case of [6], the rotation generates D3 brane charge. This is the reason why the SO (1 ,
3) isometry of the background is untouched, in spite of doing different operationsin time (boost) and in the R directions (T-dualities). As a by-product of the generationof D3 branes the dilaton is invariant under this set of operations. In other areas of Physics, solution generating techniques play a major role. For example, many of theinteresting solutions in higher dimensional gravity were constructed using solution generating algorithms. Seefor example [4] for original papers and [5] for a nice summary with an important application. It can be seen that this U-duality (sequence of T-dualities, lifts and boosts) describedabove is equivalent to a particular case of ‘rotation’ in the SU (3) structure of the mani-fold M which is characterized by a complex 3-form Ω and a 2-form J - see the papers[12] and [13]. This is another way of understanding the presence of the SO (1 ,
3) isometry:from this perspective, all the rotation ‘occurs’ in the internal M . • The rotation generates a solution that contains two free parameters, the value of thedilaton at infinity (already present in the seed solution) and the boost parameter β .In order for the final background to be dual to the Klebanov-Strassler QFT decoupled from gravity, we must take the limit β → ∞ so that the generated warp factor vanishesasymptotically. In this sense, the seed solution describes a field theory coupled to gravity.Only in a particular limit-see the discussion in [6], [13], we approach the near branesolution of [11]. • This rotation or U-duality, is a very curious operation from the viewpoint of the dual fieldtheory. Indeed, it generates (aside from the bifundamental fields) new global symmetries,like the baryonic U (1). • In the same vein: the connection between the two field theories from a geometric view-point (Field theory at strong coupling!) is just the mentioned U-duality starting fromthe D • After the first set of T-dualities described above, we have a IIA background with D2brane charges. When lifted to M-theory we have M2 branes, that after boosted generateM2 charge and an M-wave. When reduced, this generates a D0, D2 and NS H field.The generation of the H in the presence of RR three-form implies the need to generate F . See Appendix E for technical details. • We emphasized that the U-duality ( chain of T-dualities, lifts and boost) described aboveis a particular case of the rotation of J , Ω discussed in [12], [13], [14]. On the otherhand, the rotation of J , Ω relies on the background being supersymmetric, while theU-duality can be also applied to Non-SUSY backgrounds. This will play an importantrole in the rest of this paper. In this paper we will try to gain a different perspective on this solution generating technique.We will not focus on the rotation of J, Ω but rely on the U-duality description (in spite of the3atter being a particular case of the former). Our interests will be two-fold. On one hand, wewill try to get a better handle on the generating algorithm by changing it (suggesting otherrelated algorithms) and applying it to various cases. On the other hand, we will apply itto backgrounds that do not preserve SUSY. We will study the effect of the rotation on blackhole solutions, generating new non-extremal solutions with horizons and other curious featuresthat will be discussed. We will also start the study of the properties of those newly generatedsolutions.This paper is organized as follows: In sections 2 and 2.1 as a warm-up exercise we discussa chain of dualities and boost acting on non-extremal Dp branes. In section 2.1.1 we willapply this to rotate a background dual to a version of Yang-Mills. Then, in section 2.2 we willstudy a variation of this U-duality that will clarify various aspects of the coming material. InSection 3 we will U-dualize a new solution describing the non-extremal deformation of a stackof N c D5 branes wrapping a two-cycle inside the resolved conifold. We will elaborate uponvarious aspects of this particular rotation in section 3.5. Finally in Section 4 we comment ondecoupling aspects of the rotation. Various appendices complement our presentation. Theyhave been written with plenty of detail hoping that colleagues working on these topics willfind them useful. We close with a summary, conclusions and a list of possible future projects.
In this section, we will start with a simple example of the ‘rotation’ procedure. That is, asequence of T-dualities, bringing the background to a type IIA solution of the supergravityequations of motion, followed by a lift to eleven dimensions, where we will boost the config-uration. We will then reduce to IIA and T-dualize back, to what we will call the ‘rotatedbackground’. As a toy example, in this section we will rotate flat Dp branes; first in p direc-tions and then in p − q directions. We will do this in detail to appreciate the differences thisintroduces in the generated solution. In Appendix D, we will propose and analyze anotherpossible U-duality to generate new solutions. Consider backgrounds of IIA/IIB of the form, ds = H ( ρ ) − / h − h ( ρ ) dt + d~x p i + H ( ρ ) / h dρ h ( ρ ) + ρ d Ω − p i ,e φ [ initial ] = e φ ( ∞ ) H − p , ∗ F p +2 = Q Vol Ω − p ,F p +2 = − ∂ ρ A ( ρ ) dt ∧ dx ∧ .... ∧ dx p ∧ dρ (2.1)4ith, H ( ρ ) = 1 + (cid:16) L p ρ (cid:17) − p , h ( ρ ) = 1 − (cid:16) R T ρ (cid:17) − p , A ( ρ ) = αg s H ( ρ ) , (2.2)where α = ˜ QL − pp and Q = (7 − p ) ˜ Q is related to the charge of the solution (see Appendix B fordetails). If we choose R T = 0, the configuration above is typically singular (except for p = 3)and preserves 16 SUSYs. Recall that for even (odd) values of p , we are dealing with a solutionof Type IIA(B) supergravity.Now, we will ‘rotate’ this background, by first T-dualizing in the p directions, this will bringus to a IIA solution, we will lift the solution to eleven-dimensional supergravity and performa boost of rapidity β in the eleventh-direction. We will then reduce to IIA and T-dualizeback in the ~x p coordinates to obtain what we call the ‘rotated background’. Our final rotatedbackground is given by , ds = H ( ρ ) − / R ( ρ ) h − h ( ρ ) dt + d~x p i + R ( ρ ) H ( ρ ) / h dρ h ( ρ ) + ρ d Ω − p i (2.3) e φ [ final ] = e φ [ initial ] R ( ρ ) − p = e φ ( ∞ ) ( H ( ρ ) R ( ρ ) ) − p (2.4) F p +2 = − ∂ ρ a ( ρ ) dt ∧ dx ∧ .... ∧ dx p ∧ dρ (2.5)where we have defined, R = ( A ( ρ ) sinh β + cosh β ) − h ( ρ ) sinh βg s H ( ρ ) , (2.6) a = 1 R [ A ( ρ ) cosh 2 β + ( A ( ρ ) + 1) cosh β sinh β − h ( ρ )2 g s H ( ρ ) sinh(2 β )] (2.7)We have explicitly checked that the rotated background is a solution of the equations ofmotion. Notice that eq. (2.6) can be written as R = cosh β + A ( ρ ) sinh 2 β + ( R T L p ) − p sinh βH ( ρ ) g s (2.8)which makes clear that R is strictly positive. In a similar way, eq. (2.7) can be written as a = 1 HR " g s α cosh(2 β ) + (cid:18) R T L p (cid:19) − p sinh β cosh βg s + H ( ρ ) sinh β cosh β . (2.9)The metric (2.3) has the structure of a warped space, ds = H ( ρ ) − / h − h ( ρ ) dt + d~x p i + H ( ρ ) / h dρ h ( ρ ) + ρ d Ω − p i , (2.10) In Appendix A, we will present the intermediate steps of this calculation for the extremal case h = 1. H ( ρ ) = H ( ρ ) R ( ρ ) = H ( ρ ) cosh β + αg s sinh 2 β + ( R T L p ) − p sinh βg s (2.11)Note that H ( ρ ) is a harmonic function of the transverse space. Hence, after the rotation in p directions we are left with a Dp brane solution. Indeed, in terms of the new warp factor, thedilaton and gauge potential are e φ [ final ] = e φ ( ∞ ) H ( ρ ) − p ∂ ρ a ( ρ ) = (cid:20) αg s + ( R T L p ) − p tanh βg s (cid:21) ∂ ρ (cid:16) H ( ρ ) (cid:17) = ˜ α ∂ ρ (cid:16) H ( ρ ) (cid:17) , (2.12)which together with (2.10) define a Dp brane background with a β dependent RR charge. Atinfinity the warp factor asymptotes to a constant, H ( ρ ) ∼ cosh β + αg s sinh 2 β + ( R T L p ) − p sinh βg s , (2.13)and the space is asymptotically flat, as expected. However, it is interesting to note that evenif we start with the Dp branes after the decoupling limit is taken, that is H = L − pp ρ − p , we willhave in the UV that, H ( ρ ) ∼ L − pp ρ − p cosh β + αg s sinh 2 β + ( R T L p ) − p sinh βg s (2.14)So, again, the warp factor asymptotes to a constant. In other words, this rotation is takingthe configuration out of the decoupling limit or coupling the field theory modes to gravity.As is well known, before the rotation, the charge of the Dp brane solution is quantized.After the rotation the charge is,1(2 πl s ) − p Z S − p ∗ F p +2 = ˜ α ρ − p H ( ρ ) V ol S − p = Q cosh β + R T − p c p g s sinh 2 β (2.15)where c p = (7 − p ) V ol S − p (2 πl s ) − p g s . Note that, generically, the charge is not quantized. This is notunusual since the supergravity dualities involved in the rotation procedure are a symmetry ofthe supergravity equations and not of the full string theory.We will now move to study a more interesting example from the viewpoint of gauge-stringsduality. We will apply the rotation procedure to the a non-SUSY Yang-Mills-like theory, firstpresented in [17] and further studied in [18], [19].6 .1.1 Example: rotation of a dual to a Yang-Mills-like theory The original background, consists of the decoupling limit of a stack of N c D4 branes wrappinga circle with SUSY breaking boundary conditions. It was discussed with details in variouspublication, see for example [17], [18], [19]. We summarize it here (in string frame) ds = H ( ρ ) − / h − h ( ρ ) dt + d~x + f ( ρ ) dx i + H ( ρ ) / h dρ f ( ρ ) h ( ρ ) + ρ d Ω i ,e φ [ initial ] = g s H − ,F = − ∂ ρ A ( ρ ) dt ∧ dx ∧ .... ∧ dx ∧ dρ F = ∗ F (2.16)where A ( ρ ) = g s H ( ρ ) and V ol is the volume of the unit four- sphere Ω .. The functions (inthe low/zero temperature phase) are given by H = ( Lρ ) , f = 1 − ( R kk ρ ) , h = 1 , (2.17)where L = N c πg s l s and the coefficient R kk is a free parameter. In the high temperaturephase (with the coordinate x compactified) we have, H = ( Lρ ) , h = 1 − ( R T ρ ) , f = 1 (2.18)where R T is a free parameter related to the temperature of the system, T = 34 π ( L R T ) − / (2.19)Proceeding as described in the previous section we rotate the background by applying the U-duality already discussed: four T-dualities in the worldvolume coordinates, uplift to M-theory,boost, reduce to IIA and T dualize back. The rotated background is, ds = H ( ρ ) − / h − h ( ρ ) dt + d~x + f ( ρ ) dx i + H ( ρ ) / h dρ f ( ρ ) h ( ρ ) + ρ d Ω i ,e φ [ final ] = g s H − ,F = a ( ρ ) dt ∧ dx ∧ .... ∧ dx ∧ dρ (2.20)where the functions H ( ρ ) , a ( ρ ) now read H ( ρ ) = H ( ρ ) cosh β + 1 g s sinh 2 β + ( R T L ) sinh βg s (2.21) a ( ρ ) = 1 R ( ρ ) h A ( ρ ) cosh 2 β + ( A ( ρ ) + 1) cosh β sinh β − f ( ρ ) h ( ρ ) sinh(2 β )2 g s H ( ρ ) i (2.22) In [18] the authors performed a rescaling of the RR potential: C p +1 → κ µ − p π C p +1 and as a result theRR charge in their paper is measured in units of 2 π . We do not perform such rescaling here. R ( ρ ) = H ( ρ ) H ( ρ ) . As before, one can check explicitly that the background above is a solutionof the eqs. of motion, but for this we must set either R T = 0 for the low/zero temperaturephase or set R kk = 0, compactifying the x direction, in the high temperature phase.As explained in the previous section, in spite of having started with a decoupled background,after the rotation we have an asymptotically flat space, indicating that the rotation has coupledthe field theory modes to the gravitational ones. We will study now a peculiar situation, where we have Dp branes and we separate a q manifoldfrom the p + 1 dimensional world-volume. The internal manifold need not be specified. Wewill impose on our U-duality that: • after the initial T-dualities, we are in IIA background, so that this is upliftable to M-theory • that the boost generates a NS H field, or what is equivalent, that the IIA configurationafter the initial T-dualities contains an electric F .We will assume (though this need not be the case) that the internal q-manifold is a torus. Inthat case, the solution reads (in string frame as usual) , ds = H − / [ − dt + dx p − q + dσ q ] + H / [ dρ + ρ d Ω − p ] ,F p +2 = ∂ ρ ˆ Adt ∧ dx ∧ ... ∧ dx p − q ∧ dσ q ∧ dρ,e φ [ initial ] = e φ ( ∞ ) H − p . (2.23)We will proceed as follows: first we will do T dualities in the x p − q directions, this will leaveus in a IIA configuration, we will lift then to M-theory and boost. We reduce to IIA and Tdualize back, all the details are written in Appendix C. We end up this duality-chain with afinal configuration ds II,st = g tt dt + d~x p − H / B / + B / (cid:16) H − / d~σ + H / [ dρ + ρ d Ω − p ] (cid:17) ,F p +2 = ∂ ρ ˆ A (cosh β + a t sinh β ) ∧ dρ ∧ dσ ∧ dσ ∧ dx ∧ .... ∧ dx p − ,H = sinh β∂ ρ ˆ Adρ ∧ dσ ∧ dσ ,F p = ∂ ρ ( a t ) dt ∧ dρ ∧ dx ∧ .... ∧ dx p − ,e φ [ final ] = B − p H − p (2.24) we are dealing here with the Lorentz invariant case here. Working with a non-extremal factor is straight-forward. A = H − / [ H sinh β − cosh β ] , B = H − / [ H cosh β − sinh β ] ,C = 2 H − / sinh β cosh β (1 − H ) , g tt = 4 AB − C √ B ,a t = C B = sinh β cosh β (1 − H ) H cosh β − sinh β , (2.25)Various aspects are worth emphasizing in this final background. Notice that: • the metric is dual to a field theory with SO (1 , p −
2) isometry, since(4 AB − C ) H / = − → g tt = − g x i x i (2.26) • for p = 5 the dilaton is invariant under the whole U-duality procedure, since e φ [ initial ] = H − p = e φ [ final ] = B − p H − p (2.27) • for p = 5 the procedure is generating charge of D3 brane (represented by the F p =5 ) wherethe D3 branes are extended in the p − q directions. This is the reason why the dilatondoes not change as the D3 branes do not couple to it. The field H is also generated.In following sections, we will study a particular case of this U-duality, for a situation in whicha set of D5 branes wrap a curved two-manifold (but we also have a black hole in the metric,breaking the SO (1 ,
3) isometry), the results are qualitatively the same. One should emphasizethat there is yet another way of generating NS three-form fields, that is basically starting fromNS five branes in IIA wrapping a three-cycle, see [20] for details.Also, notice that we have imposed that after the first set of T-dualities, the backgroundis solution of IIA Supergravity (the conditions for this to happen are discussed in AppendixC). Were this not the case, we present in Appendix D another possible solution generatingtechnique with some applications.Now, we will move to study an interesting application of the U-dualities discussed above. N = 1 SUSY Solutions
As anticipated, in this section we will apply the chain of dualities, lift and boost proposedin [6], to generate a new solution starting from a non-extremal solution in Type IIB. Theinterest of the original (‘seed’) solution is that it was argued to be dual to a field theory withminimal SUSY in four dimensions. The field theory was studied (at weak coupling) in [16] andis basically N = 1 Super-Yang-Mills plus a sets of (massive) KK vector and chiral superfields9hat UV complete the dynamics. This system was well studied and various string duals areknown (particular solutions describing the field theory at strong coupling, with VEV’s andcertain operators deforming the Lagrangian). Let us briefly describe the general form ofthe string dual. The ‘seed’ background describes the backreaction of a set of N c D5 braneswrapping a two-cycle inside the resolved conifold. It consists of a metric, dilaton φ ( ρ ) and RRthree-form F and, in string frame, it reads (the coordinates used are [ t, ~x, ρ, θ, ϕ, ˜ θ, ˜ ϕ, ψ ]), ds = e φ ( ρ ) h dx , + e k ( ρ ) dρ + e h ( ρ ) ( dθ + sin θdϕ ) ++ e g ( ρ ) (cid:0) (˜ ω + a ( ρ ) dθ ) + (˜ ω − a ( ρ ) sin θdϕ ) (cid:1) + e k ( ρ ) ω + cos θdϕ ) i ,F (3) = N c " − (˜ ω + b ( ρ ) dθ ) ∧ (˜ ω − b ( ρ ) sin θdϕ ) ∧ (˜ ω + cos θdϕ ) + b dρ ∧ ( − dθ ∧ ˜ ω + sin θdϕ ∧ ˜ ω ) + (1 − b ( ρ ) ) sin θdθ ∧ dϕ ∧ ˜ ω . (3.28)where ˜ ω i are the left-invariant forms of SU (2)˜ ω = cos ψd ˜ θ + sin ψ sin ˜ θd ˜ ϕ, ˜ ω = − sin ψd ˜ θ + cos ψ sin ˜ θd ˜ ϕ , ˜ ω = dψ + cos ˜ θd ˜ ϕ. (3.29)This supersymmetric system was carefully studied in a series of papers [9], [10]; where it wasshown that there is a combination of background functions (basically a ‘change of basis’) thatmove from a set of coupled BPS equations to a decoupled one that one can solve-up to onefunction, that we will call P ( ρ ). We will not insist much with this formalism here and referthe interested reader to the original work [9], [10] .We will be more restrictive and for the purposes of this paper, we will study solutions wherethe functions a ( ρ ) = b ( ρ ) = 0 in the background of eq.(3.28). This is just in order to makeour numerics simpler and illustrate the points we want to make. If we wanted to constructa black hole solution showing explicitly the transition between R-symmetry breaking and itsrestoration, we should then work without this restriction. Since in all known solutions theasymptotics of the functions a ( ρ ) ∼ b ( ρ ) ∼ e − ρ our asymptotic results will be qualitativelycorrect, but near the black hole horizon there could be important differences.We will then proceed as follows: first we will propose a background including a black holewith the restrictions mentioned above ( a ( ρ ) = b ( ρ ) = 0). We will then pass it through thesolution generating machine. This will produce a new background, now including also F , H ,that we will write explicitly (we have checked that the equations of motion before and afterthe U-duality are the same). We will be explicit about the asymptotics of each of the functions All this formalism was also applied to the case in which one also adds fundamental matter to the dualQFT or sources to the background above. We refer the interested reader to the review [21]. Z N → Z . In this section we consider the background presented in equation (3.28) with the restrictions a = b = 0 but including the non-extremality factors. In string frame we have, ds IIB,s = e φ h − h ( ρ ) dt + dx + dx + dx i + ds ,s ,ds = e φ h e k s ( ρ ) dρ + e k ω + cos θdϕ ) + e q ( dθ + sin θdϕ ) + e g d ˜ θ + sin ˜ θd ˜ ϕ ) i ,F (3) = N c " − ˜ ω ∧ ˜ ω + sin θdθ ∧ dϕ ∧ (˜ ω + cos θdϕ ) = α N c w , (3.30)where h ( ρ ) , s ( ρ ) are the non-extremality functions. In the following, we will set s ( ρ ) = h ( ρ )which is simply a choice of parametrization; this implies that (on a particular solution) otherbackground functions { q ( ρ ) , g ( ρ ) , k ( ρ ) , φ ( ρ ) } need not take the same values as in the SUSYbackground.The above Ansatz might be familiar to some readers. It is worth highlighting a key differencewith previous work. Since our goal is to ‘rotate’ this solution we look for solutions withstabilized dilaton. Namely, the typical asymptotic value of the dilaton inherited from the D5(or NS5 brane) solution is linear [32], more precisely [23] [25], e φ ∼ e ρ ρ − / (3.31)We are looking for a dilaton that stabilizes at infinity, that is, which asymptotically behavesas: e φ ∼ O (cid:0) e − / ρ (cid:1) . (3.32)Our solution is qualitatively characterized by two parameters: one describing the nonextremal-ity of the solution and the other the speed at which the dilaton gets stabilized.After applying the solution generating technique, we end up with a new Type IIB back-ground containing F , H aside from the fields originally present in eq.(3.30) (all details are11ritten in the Appendix E), ds IIB,st = H − / h − hdt + dx + dx + dx i + e φ H / h e k dρ h ( ρ ) + e k ω + cos θdϕ ) ++ e q ( dθ + sin θdϕ ) + e g d ˜ θ + sin ˜ θd ˜ ϕ ) i ,H / = cosh β q e − φ − h ( ρ ) tanh β, φ new = φ ( ρ ) ,F = ( α N c βw , H = −√ h ( α N c β e φ ∗ w F = − tanh β (1 + ∗ )vol ∧ d ( hH ) . (3.33)In the following, we will focus in the limit β → ∞ , this is the field theory-limit where thewarp factors vanish at infinity. We will then rescale˜ N = N cosh β, x i → p cosh β p ˜ N α x. (3.34)With all the above the β → ∞ limits are finite and the solution is given by ds IIB,st = ˜
N α h ˜ H − / ( − hdt + dx i dx i ) + e φ ˜ H / ds i ,F = α ˜ N w , H = −√ h α ˜ N e φ ∗ w F = − ( α ˜ N ) (1 + ∗ )vol ∧ d ( h ˜ H ) (3.35)with ˜ H / = p e − φ − h ( ρ ) (3.36)It is worth making a few observations: • We have two new classes of solutions. The equations of motion before and after the‘rotation’ are the same, thus a solution of the system of equations (3.30) furnishes asolution of the form (3.33)-(3.35). • The non-extremality factor have made its way into the RR five form and the warp factorin eq.(3.36), but the dilaton and the three forms are unchanged from the extremal case.Notice that the factor of √ h in the NS field H is canceled by the self dual in sixdimensions ∗ w . See Appendix E for full details. • The black hole in the seed solution will likely have negative specific heat but its dilatonis stabilized which is a crucial difference with solutions considered previously. We willdiscuss what is the behavior of the solution after the rotation.In the following section we will explicitly describe the asymptotic behavior of the solution aswell as its numerical presentation in the background given by eq.(3.30).12 .2 Asymptotics
We will proceed to study the asymptotics of the equations of motion and solve them numeri-cally to obtain the new non-extremal seed solution (3.30).
A large ρ expansion of the functions ( k, q, g, h, φ ), that solve the equations of motion up toterms decaying faster than e − ρ is h ( ρ ) = e − x ( ρ ) ∼ C e − ρ ,e q ( ρ ) ∼ N c ρ −
1) + c e ρ , e g ( ρ ) ∼ N c − ρ ) + c e ρ ,e k ( ρ ) ∼ c e ρ − N c c (25 − ρ + 16 ρ ) e − ρ , e φ − φ ∼ N c c (1 − ρ ) e − ρ . (3.37)There are two parameters in the expansion above. The parameter C characterizes the non-extremal behavior. The parameter c plays a quite important role. It is one of the integrationconstants of the BPS eqs. As an expansion in inverse powers of this constant it is possibleto write a solution for the background in eq.(3.28)-see for example [10], [22]-such that whenpassed through this solution generating technique, it results in the Baryonic Branch dualsolution of [7]. Physically (and in the background before the rotation), the constant c isdescribing the coupling of the dynamics on the five-brane to gravity and ultimately to the fullstring theory, it takes the configuration out of its ‘near brane’ limit and can be thought of asindicating the insertion of an irrelevant operator in the QFT lagrangian. Near the horizon, ρ = ρ h , we expand the equations of motion in power series up to order 7.We have found, e − x ( ρ ) = x ( ρ − ρ h ) + x ( ρ − ρ h ) + x ( ρ − ρ h ) + · · · + x ( ρ − ρ h ) e q ( ρ ) = q + q ( ρ − ρ h ) + q ( ρ − ρ h ) + · · · + q ( ρ − ρ h ) e g ( ρ ) = g + h ( ρ − ρ h ) + g ( ρ − ρ h ) + · · · + g ( ρ − ρ h ) (3.38) e k ( ρ ) = k + k ( ρ − ρ h ) + k ( ρ − ρ h ) + · · · + k ( ρ − ρ h ) e ρ ) = f + f ( ρ − ρ h ) + f ( ρ − ρ h ) + · · · + f ( ρ − ρ h ) (3.39)Demanding that the expansions in eq.(3.38) satisfy the equations of motion of the systemwe determine x , ..., h ..., g ..., k ..., f ... in terms of x , h , g , k , f and thus, there are only 5 Note that the UV expansion for h ( ρ ) = e − x could have started with a lower order term such as e − ρ , seeSection 4.1 for a discussion on the choice of parameters. h 3.3 3.4 3.5 3.6 3.7 3.8 3.9 401020304050 r Figure 1:
Solution before the rotation for c = 1 . C = − e q , e k , e g and h − respectively. rh 2.0 2.5 3.0 3.5 4.0020406080100120140 r Figure 2:
Solution before the rotation for c = 125 . C = −
89. The blue, dashed,green ad orange lines represent e q , e k , e g and h − respectively. independent parameters coming from the expansion at the horizon. Furthermore, x is relatedto the non-extremality parameter α -see Appendix F for the definitions, α = − x √ f g h The strategy we will follow is to numerically integrate back from infinity the equations ofmotion in Appendix F, using as boundary conditions the expansion in the UV-eq.(3.37)-andrequire that at the horizon this numerical solution matches the expansion (3.38) and its firstderivatives -up to seventh order. Doing this for a given set of values ( c, C ), determines thefree parameters ( x , h , g , k , f ). Note that not for every pair of ( c, C ) there is a blackhole solution. For some values of ( c, C ) there will not be a horizon and for others theremight be naked singularities outside the horizon. The study of the this two-parameter familyof solutions for the full ( c, C ) parameter space is an interesting question that we will notaddress here. We concentrate on finding particular values of ( c, C ) that will produce blackhole solutions. Needless to say, finding a solution before the rotation guarantees that we willhave a solution after the rotation. Examples of numerical solutions are shown in Figure 1 and2 and the respective dilatons in Figures 3 and 4. As mentioned above we integrate back from infinity and match with the expansion at thehorizon. In order to do so we define a “mismatch” function evaluated at a point ρ close to14 h 3.5 3.75 4.00.9960.9970.9980.9991.000 r (cid:227) (cid:70) Figure 3:
Dilaton, e for c = 1 . , C = − . rh 2.0 2.5 3.0 3.5 4.00.999950.999960.999970.999980.999991.00000 r (cid:227) (cid:70) Figure 4:
Dilaton, e for c = 125 . , C = − the horizon, ρ = ρ h + 0 . m ( ρ ) =[ g sh ( ρ ) − g num ( ρ )] + [ g sh ( ρ ) − g num ( ρ )] + [ h sh ( ρ ) − h num ( ρ )] +[ h sh ( ρ ) − h num ( ρ )] + [ k sh ( ρ ) − k num ( ρ )] + [ k sh ( ρ ) − k num ( ρ )] +[Φ sh ( ρ ) − Φ num ( ρ )] + [Φ sh ( ρ ) − Φ num ( ρ )] . (3.41)The subscript num denotes the numerical solution obtained by integrating back from infinityand sh denotes the series expansion (3.38) -evaluated at ρ . Recall that the expansion atthe horizon depends on five free parameters, ( x , h , g , k , f ). For a given set of ( c, C ) weminimize the mismatch function to determine ( x , h , g , k , f ). The parameters at infinity( c, C ) are adjusted so that m ( ρ ) < × − . All the numerical procedures where done inMathematica using WorkingPrecission 40. We also checked that the constraint T + U = 0(see appendix F) remains numerically negligible throughout the domain. Once we have obtained a numerical solution for the non-extremal background (3.30) we caneasily generate the rotated solution as summarized in eq.(3.35) and outlined in the AppendixE. Before presenting the numerical results let us look at the large ρ asymptotics. Using the UVexpansions of the seed solution (3.37), one can obtain the black hole metric (3.35) asymptotics15 fter the rotation, − g tt ∼ e / ρ A ( ρ ) − e − / ρ c √ A ( ρ ) [256 c C + 27(1 − ρ ) + 96 c C ( − ρ )] + · · · ,g xx ∼ e / ρ A ( ρ ) − e − / ρ c √ A ( ρ ) [27(1 − ρ ) ] + · · · ,g ρρ ∼ c A ( ρ ) + · · · , g θθ ∼ c A ( ρ ) + e − / ρ A ( ρ )(2 ρ −
1) + · · · ,g ψψ ∼ c A ( ρ ) + · · · , g ˜ θ ˜ θ ∼ c A ( ρ ) − e − / ρ A ( ρ )(2 ρ −
1) + · · · . (3.42)Where we have defined the quantity c √ A ( ρ ) ≡ p ρ − C c − . (3.43)Using a variable u = e ρ/ the metric can be written -to leading order- as, ds = u A ( u ) ( dx i dx i ) + 3 cA ( u )2 ( du u + ds T ) + · · · O ( u − ) . (3.44)Thus, the non-extremal solution found after the rotation has the asymptotics of the Klebanov-Strassler background [8]. We point out that the reason we get precisely Klebanov-Tseytlintype of asymptotics lies in our precise arrangements of coefficients in the seed solution. Namelywe arranged for a seed solution with no linear dilaton and with a dilaton behavior that, inthe field theory limit, eliminates the leading term in the harmonic function given in equation(3.36).With our careful choice of asymptotic conditions for the seed solution of D5 branes wehave managed to construct black holes in asymptotically KT backgrounds. Our solution isdifferent from those obtained in the traditional approach to black holes in asymptotically KTbackgrounds (see, for example, [33]). One glaring difference is the form of the RR fluxes. Inparticular the presence of the non-extremality parameter in F and in the warp factor H ( ρ )is completely novel.In Figures 5-8 we present the numerical plots of the diffent metric elements after the rotationfor two sets of parameters. In Figures 9 and 10 we see how the angular part of the solutionapproaches the T , metric for the two representative solutions. Let us clarify the expectations of the form of the background that arise from field theoryconsiderations. In the previous section we obtained a solution largely characterized by twoparameters ( c, C ). After the rotation, the interpretation of c has been spelled out explicitly16 h 2 2.5 3 3.5 4.0100015002000250030003500 Figure 5:
The g rr metric element for a so-lution after the rotation, c = 125 . , C − rh 3.4 3.6 3.8 4.00500100015002000 Figure 6:
The g rr metric element for a solu-tion after the rotation, c = 1 . , C = − rh 2 2.5 3 3.5 4.005101520 Figure 7: g xx and g tt for a solution after therotation, c = 125 . , C = − rh 3.4 3.6 3.8 4.00.00.51.01.52.02.53.0 Figure 8: g xx and g tt for a solution after therotation, c = 1 . , C = − Figure 9:
Angular part of the metric afterthe rotation. We plot g θθ , g ˜ θ ˜ θ and g ψψ . So-lution with c = 125 , C = − Figure 10:
Angular part of the metric afterthe rotation. We plot g θθ , g ˜ θ ˜ θ and g ψψ . So-lution with c = 1 . , C = −
17n [15], [6] and [13]. It is related to the vev of the baryon operator. The other parameterpresent in the solution is C , its interpretation coincides roughly with the temperature of thesolution in the field theory.We will present and exhaustive analysis of these classes of solutions elsewhere and will limitour attention here to a few main points. The main deterrent in presenting a final descriptionof the thermodynamics is the need to explore a large parameter space of the solutions as wellas the difficulty in matching data at asymptotic infinity with data near the horizon. Theanalysis is numerically costly but we emphasize that these are not difficulties of principle.Our goal in this work is to explicitly show, through numerical analysis, the existence of theconjectured solutions.Let us show that the thermodynamic analysis can, in principle, be done explicitly for thesolutions we have. For the temperature, we use the standard approach of looking at the Eu-clidean section and imposing the absence of conical singularities to determine the temperatureof the supergravity backgrounds. Basically, for a metric with Euclidean section of the form ds = f dτ + dρ g + . . . , (3.45)we find a temperature equal to T = 14 π p f g , (3.46)where prime represents derivative with respect to the radial coordinate and the above expres-sion is evaluated at the horizon. Interestingly, the temperature in both backgrounds, that is,before and after the rotation, is the same. In terms of the near horizon data we have T = 14 π x √ k . (3.47)It is worth noting that this is the temperature at the horizon. In the case of asymptoticallyflat backgrounds (before the rotation) we have the option of computing the temperature asseen by an observer at infinity. We refer the reader to [23] for a detailed discussion of thisprocedure in the D5 or NS5 background. The entropy is computed, as usual, as the area ofthe horizon. The most complicated quantity to compute is the free energy which is requiredto compute, for example, the specific heat. For the free energy we need to evaluate the action.In the case of asymptotically flat solutions for D5/NS5 branes we can essentially follow [23].A very preliminary analysis seems to confirm our intuition about the specific heat of suchsolutions. Namely, evaluation of the action suggests that before the rotation the solutionsobtained have negative specific heat. A precise evaluation of the action after the rotationis more subtle as naive evaluation yields divergent terms. It is likely that a more accurateapproach requires some sort of holographic renormalization or substraction along the lines of[33]. Preliminary evaluation suggests a positive specific heat for the rotated solutions . Dario Martelli suggested that along the lines of the first two papers in [3], some thermodynamical quantitiesshould be invariant under T-dualities, shifts of coordinates and boosts. See also [34]. About decoupling limits
It is worth discussing a bit about decoupling limits. Let us do this in the extremal case, theextension to the cases with non extremal factors follows from what we write here. The firstpoint to emphasize is that when gravity modes are coupled to a generic D-brane configuration,the warp factors asymptote to a constant. An example of this is what happens to the flat Dpbranes, with H = 1 + ( Qρ ) − p .We will compare warp factors in the examples studied in the previous sections. It is enoughto focus on the expressions of the metric component g tt . In all the examples studied we found g tt = 4 AB − C √ B . (4.48)Indeed, in the case in which we rotate a flat Dp-brane, we have eq.(A.6) with
A, B, C given ineq.(A.5). In the case of the rotation in p − q directions the results for A, B, C are written ineq.(2.25) and in the case of the wrapped D5’s are given in eq.(E.9). Again, for the point wewish to emphasize, we will focus on the extremal cases, when the rotation preserves SO (1 , p ).So, in these cases we have , g tt, ( p − q ) = − β p H − tanh β , g tt,p = − √ H q H ( ˆ A tanh β − − tanh β ,g tt,D = − β p e − φ − tanh β (4.49)The case of the rotation in p directions shows that the induced charge of D0 brane (after thefirst set of T-dualities) as shown by eq.(A.2), enters in the new warp factor via the one-formpotential A = ˆ Adt . This makes the decoupling of the final configuration more unclear andtypically, if we start with ‘decoupled’ Dp branes (that is H ∼ ρ p − ) we will generate a non-decoupled configuration, with an asymptotically constant warp factor as discussed aroundeq.(2.13). Of course, we can try to take the decoupling limit in the final configuration.On the other hand the cases of the wrapped D5’s and the rotation in p − q directions(that share the fact that a D0 charge is induced by the boost only) are such that we must start with a non-decoupled configuration in order to avoid a generated background with asingularity or end of the space in the UV (zero warp factor at some finite value of the radialcoordinate). Indeed, if H = 1 + ( Qρ ) − p we choose a particular value of the boost parameter β → ∞ , together with a particular choice of the value of e φ ( ∞ ) = 1 in the D5’s case) to obtaina background decoupled from gravity . Gregory Giecold correctly pointed out that in the non-extremal cases the non-extremality factor h ( ρ )enters as expected, for example g tt,D = − h cosh β √ e − φ − h tanh β . We thank him for this and various othercorrect comments. One of the interesting observations in [6], is that in the case of the D5’s the solution described in the .1 Choice of Parameters for N = 1 Black Holes
Let us discuss in more detail the choice of parameters in the solution. This is a particularlysensitive question from the numerical and from the physical points of view. We note that apriori there are ten parameters in our solution as corresponds to a system of five second orderordinary differential equations. The constraint lowers this number by one. At this point thesimplest route for us is to notice that we are looking for a particular solution that asymptotesto the extremal supersymmetric one found in [10],[13] (see, in particular, appendix B there).In that context most of our constants get fixed, as we require the same UV behavior; we usethe expressions in that solution to fix six constants. Of the three constants left, one gets fixedby the asymptotic behavior of the non-extremality function, that is, we choose h ( ρ ) ∼ − C e − ρ . (4.50)The leading coefficient guarantees that we will have, after rotation, a solution with KT asymp-totics. The key observation to be made is that, in principle, there is a mode allowed that wehave explicitly set to zero. Namely, another solution to the equations of motion in the asymp-totic regime could be h ( ρ ) ∼ C e − ρ − C e − ρ + . . . . (4.51)The above expression should be thought of schematically as the solution to a second orderdifferential equation for h ( ρ ) which is defined asymptotically by two parameters. This C is one of the three constants that we fix and therefore we remain with two constants thatwe called ( c, C ). The parameter c has the same meaning discussed in [10],[13] and C isrelated to the temperature. Thus, the condition that the solution after the rotation has KTasymptotics plus a condition on the non-extremality function determine the values of theintegration constants.It is worth noting that selecting the right constants in the D3/D5 picture is more cumber-some as there are more parameters in the generic D3/D5 approach to black holes with KTasymptotics. For example, our solution does not allow for a generic mode that will lead toa Freund-Rubin form for F . As has been repeatedly emphasized, our F is related to thenon-extremality function and deforms the typical term proportional to the volume 5-form.Let us explicitly visualize the difficulty in choosing the asymptotic expansion from the D3/D5point of view in another example. The choice C = 0 roughly corresponds to a dimension3 operator in the D3/D5 picture. This can be seen easily by going to the radial coordinate u = exp(2 ρ/ papers [10], [22] is such that the dilaton asymptotes to a (tunable) constant and the subleading terms areprecisely those needed to generate the warp factor of Klebanov-Strassler Conclusions and Future Directions
We present here some concluding remarks together with some possible future projects tocomplement and refine the material in this paper.In this work we have elaborated on the U-duality proposed in [6]. In that case, it was usedto connect solutions describing wrapped D5 branes with the family of solutions describing thestrong coupling dynamics of the baryonic branch of the Klebanov-Strassler field theory. In ourcase, we have used this U-duality to construct a new family of solutions describing the finitetemperature phase of this baryonic branch. But before doing so, we explored the U-duality(that, following custom, we called rotation) in more generality, as a solution generating tech-nique. We applied it to flat Dp branes in two different circumstances so that we appreciatedthe subtle differences making the choice of dualities in [6] so special.Once we developed this technology, we applied it to rotate the Witten model for a Yang-Millsdual [17]. All this represents new material and it would be interesting to find an applicationfor it, hopefully in the context of duals to Yang-Mills or QCD-like field theories.In section 3, we discussed various new things: first a class of non-extremal solutions describ-ing five branes wrapping a two-cycle inside the resolved conifold. The main feature of thesenon-extremal backgrounds is that the dilaton is asymptotically stabilized, which we inter-preted as coupling the dual field theory to gravity modes. Then, we applied the U-duality tothis solution, to find a new background of IIB that describes a non-extremal deformation of theKlebanov-Tseytlin solution ( decoupled from gravity or stringy modes). We briefly commentedon the associated thermodynamic properties of this new family of backgrounds. Finally, wediscussed various aspects of this coupling/decoupling from gravity modes in section 4.Let us now list a set of projects that complement the material presented here and that mayimprove the understanding of the topic: • It is important to sort out the details of the thermodynamics after the rotation in thesolutions presented in section 3. Indeed, as we indicated in section 3.5, this poses anumerical problem, but certainly does not amount to a conceptual obstruction. Theresult for quantities like the specific heat or the free energy will be of great interest.Notice that this presents a novel way to approach the finite temperature phase of theKlebanov-Strassler field theory. Comparison with the solutions of [38] would be inter-esting. • More conceptually; it would be nice to better understand the role played by the D0branes in this solution generating technique. Indeed, notice the differences between theexamples analyzed in section 2 and the one in section 2.1, all due to the presence ofD0 branes before the boost that translates into the different final results. In the samevein, it would be good to have a better understanding on the reasons for the equivalencebetween the U-duality explained here and the rotation of the J , Ω forms as described21n [12], [13]. • It would be of very much interest to extend our non-extremal solution to a background ofthe form in eq.(3.28) where the restriction a ( ρ ) = b ( ρ ) = 0 is not applied. Indeed, findingsuch a non-extremal solution would, after the application of the U-duality, produce asolution dual to a non-SUSY KS-like field theory in the baryonic branch. It may proveinteresting to compare this new solution with that presented in the papers [35]. Thedifferent nature of the non-SUSY deformation will certainly reflect in different fieldtheory aspects. • It would be nice to use the solution presented in section 3 as a base to produce anon-SUSY deformation along the line proposed recently in [36]. The authors of [36]are finding SUSY breaking backgrounds by adding anti-D3 branes. Their solutionshave singularities (and the problem is if such a singularity is physically acceptable).Proceeding as with the solution in this paper may set us on a different branch of solutions. • It would be interesting to find a Physics application of the new backgrounds of section2.1.1. • It would be nice to extend the treatment here to the case in which the seed backgroundcontains N f ∼ N c flavors. This would be following the lead of the papers [24], [25]. Thesubtle point would be how to rotate such solutions (in other words, how the sourcesget represented in M-theory, see [37]. Comparison with the solutions of [38] would beinteresting) • We have presented two numerical solutions as examples of the type of backgroundsobtained. These examples correspond to specific values of the UV parameteers. We didnot attempt a study of the full family of solutions in the whole parameter space. Thisis an interesting question worth investigating.We hope to address some of these problems in the near future.
Acknowledgments:
We would like to thank various colleagues for the input that allowed us to better understandand present the results of this paper. We would like to single out Eloy Ay´on-Beato, Iosif Bena,Alex Buchel, Pau Figueras, Gregory Giecold, Mariana Gra˜na, Prem Kumar, Dario Martelli,Ioannis Papadimitriou, Michela Petrini, Dori Reichmann and C´esar Terrero-Escalante. Veryspecial thanks to Gregory Giecold for a very careful reading of the manuscript. E.C. thanksthe Theory Group at the University of Texas at Austin for hospitality.22his work was supported in part by the National Science Foundation under Grant Num-bers PHY-0969020 and PHY-0455649, the US Department of Energy under grant DE-FG02-95ER40899 and CONACyT grant CB-2008-01-104649.
A Appendix: U-duality in the case of flat Dp-branes
In this appendix we will give a sketchy derivation of the U-duality chain, in the case in whichwe rotate flat Dp-branes. We will not consider the non-extremal case, this can be done easilyand the result of doing so is included in the first section of the paper. Here we want just toemphasize some points.We start from the SUSY configuration in type II supergravity (string frame), ds = H − / ( − dt + dx p ) + H / ( dρ + ρ d Ω − p ) ,e φ = H − p , F p +2 = ∂ ρ ˆ Adt ∧ dx ∧ ...dx p ∧ dρ (A.1)where ˆ A = − A in the notation of Section 2. Following standard convention we define thecharge of F − p as positive. The sign of F p +2 will then depend on whether we considerMinkowski or Euclidean worldvolume on the D p brane. We apply T-duality in the x p di-rections, ds = − H − / dt + H / ( dx p + dρ + ρ d Ω − p ) ,e φ = H , F = ∂ ρ ˆ Adt ∧ dρ (A.2)We lift this to M-theory, using e φ = H , ds = H ( dx + ˆ Adt ) − H − dt + dM , dM = ( dx p + dρ + ρ d Ω − p ) (A.3)we now boost in the t − x direction with rapidity βdt → cosh βdt − sinh βdx , dx → − sinh βdt + cosh βdx , (A.4)to get ds = Adt + Bdx + Cdtdx + dM ,A = H − [ H ( ˆ A cosh β − sinh β ) − cosh β ] B = H − [ H ( ˆ A sinh β − cosh β ) − sinh β ] C = − H − [ H ( ˆ A cosh β − sinh β )( ˆ A sinh β − cosh β ) − cosh β sinh β ] (A.5)and we prepare the metric for reduction back to IIA as ds = B ( dx − a t dt ) + B − / h g tt dt + B / dM i ,a t = − C B , g tt = 4 AB − C √ B (A.6)23e now write the IIA configuration as, ds IIA = g tt dt + B / ( dx p + dρ + ρ d Ω − p ) , e φ = B / , F = ∂ ρ a t dt ∧ dρ (A.7)and we T-dualize back in the x p directions to get ds II = g tt dt + B − / dx p + B / ( dρ + ρ d Ω − p ) ,e φ = B − p , F p +2 = ∂ ρ a t dt ∧ dρ ∧ dx .... ∧ dx p (A.8)We see that the SO (1 , p ) invariance is preserved, as g tt √ B = − B Appendix: Black Dp branes
In this appendix we gather some useful facts about black Dp-brane solutions [26],[27],[28].The metric in string frame is, ds = − f + ( r ) p f − ( r ) dt + p f − ( r ) p X i =1 dx i dx i + f − ( r ) − − − p − p f + ( r ) dr + r f − ( r ) − − p − p d Ω − p (B.1)where f ± ( r ) = 1 − (cid:16) r ± r (cid:17) − p (B.2)The mass per unit volume M and the R-R charge N are M = 1(7 − p )(2 π ) d p l P ((8 − p ) r − p + − r − p − ) , N = c p ( r + r − ) − p , (B.3)where l P is the ten dimensional Planck length l P = g / s l s and c p , d p are numerical factors, d p = 2 − p π − p Γ (cid:18) − p (cid:19) c p = 1 d p g s l − ps = (7 − p ) V ol S − p (2 πl s ) − p g s (B.4)The background also has a nontrivial dilaton and a R-R flux, e − = g − s f − ( r ) − p − πl s ) − p Z S − p ∗ F p +2 = N. (B.5)The Einstein frame metric ( g E µν = p g s e − Φ g st µν ) has a horizon at r = r + and, for p ≤
6, asingularity at r = r − . If r + > r − the singularity is covered by the horizon and the solution isa black hole. In the extremal case, r + = r − , the space is singular except for p = 3.24et us define a new coordinate, ρ − p = r − p − r − p − . This change of variables transforms (B.1) to a more familiar form, ds = H ( ρ ) − − h ( ρ ) dt + p X i =1 dx i dx i ! + H ( ρ ) (cid:18) dρ h ( ρ ) + ρ d Ω − p (cid:19) (B.6)where H ( ρ ) = 1 + r − p − ρ − p h ( ρ ) = 1 − r − p + − r − p − ρ − p (B.7)Throughout the paper we use the notation, L − pp ≡ r − p − and R − pT ≡ r − p + − r − p − . Note that using (B.3) one can also write L − pp = ˜ Q s (cid:18) R − pT Q (cid:19) − R − pT Q = ˜ Q α p . (B.8)where ˜ Q = Nc p and α p = (1 + ( R − pT Q ) ) / − R − pT Q .¿From (B.3) we can see that the parameter R T is related to the energy per unit volumeabove extremality (cid:15) , ∆ M = (cid:15) = r − p + − r − p − (2 π ) d p l P = R T − p (2 π ) d p l P . (B.9)Note that in the decoupling limit ( α →
0, energies fixed) (cid:15) remains fixed and α p → r + = r − , we have R T = 0 and L − pp = ˜ Q = N d p g s l − ps . C Appendix: Some details on the U-duality
Here we provide details for the derivations of eq.(2.24). We started with ds = H − / [ − dt + dx p − q + dσ q ] + H / [ dρ + ρ d Ω − p ] ,F p +2 = ∂ ρ ˆ Adt ∧ dx ∧ ... ∧ dx p − q ∧ dσ q ∧ dρ,e φ [ initial ] = e φ ( ∞ ) H − p . (C.1)25he we perform the U-duality described in the text below eq.(2.23) ds = H − / [ − dt + dσ q ] + H / [ dρ + ρ d Ω − p + dx p − q ] ,F q +2 = ∂ r ˆ Adt ∧ dσ q ∧ dρ,e φ = e φ ( ∞ ) H − q . (C.2)Now, we need this to be a configuration in Type IIA (in order to lift this to M-theory). Wewill also want to impose that when lifted to M-theory this will produce a four-form field G .If this is the case, we must have that either q = 2 or that q = 4 (the cases of q = 0 , q = 6are analogous to what we analyzed in the first section). The case q = 2, on which we willelaborate upon below, has the peculiarity that the reduction from eleven dimensions back toIIA will generate a NS H field, proportional to the boost. Let us see this in detail. To beginwith, we will lift to eleven dimensions the configuration in eq.(C.2). ds = H / dx + H − / [ − dt + d~σ ] + H / [ dρ + ρ d Ω − p + d~x p − ] ,G = ∂ ρ ˆ Adt ∧ dρ ∧ dσ ∧ dσ (C.3)and now we boost with rapidity βdt → cosh βdt − sinh βdx , dx → − sinh βdt + cosh βdx (C.4)and we can rewrite the configuration after the boost as, ds = Adt + Bdx + Cdtdx + dM ,G = ∂ ρ ˆ A (cosh βdt − sinh βdx ) ∧ dρ ∧ dσ ∧ dσ ,dM = H − / d~σ + H / [ dρ + ρ d Ω − p + d~x p − ] ,A = H − / [ H sinh β − cosh β ] ,B = H − / [ H cosh β − sinh β ] ,C = 2 H − / sinh β cosh β (1 − H ) (C.5)when we reduce this to IIA we get, ds IIA,st = g tt dt + B / (cid:16) H − / d~σ + H / [ dρ + ρ d Ω − p + d~x p − ] (cid:17) ,e φ [ A ] = B / , g tt = 4 AB − C √ B ,F = ∂ ρ ˆ A (cosh β + a t sinh β ) dt ∧ dρ ∧ dσ ∧ dσ ,H = sinh β∂ ρ ˆ Adρ ∧ dσ ∧ dσ ,F = ∂ ρ ( a t ) dt ∧ dρ, a t = C B = sinh β cosh β (1 − H ) H cosh β − sinh β , (C.6)26e see that we have generated a NS magnetic field. Finally, we T-dualize back in the ~x p − directions, to get ds II,st = g tt dt + d~x p − H / B / + B / (cid:16) H − / d~σ + H / [ dρ + ρ d Ω − p ] (cid:17) ,F p +2 = ∂ ρ ˆ A (cosh β + a t sinh β ) ∧ dρ ∧ dσ ∧ dσ ∧ dx ∧ .... ∧ dx p − ,H = sinh β∂ ρ ˆ Adρ ∧ dσ ∧ dσ ,F p = ∂ ρ ( a t ) dt ∧ dρ ∧ dx ∧ .... ∧ dx p − ,e φ [ final ] = B − p H − p . (C.7)This completes our derivation of eq.(2.24). D Appendix: Another solution generating algorithm
We studied two different ‘solution generating techniques’ and applied them to different back-grounds. All these ‘algorithms’ were starting with a background solution to the Type II (Aor B) equations of motion, applying a number of T-dualities that would transform the back-ground into a solution for Type IIA supergravity. Then we lifted this to eleven dimensions,where a boost was applied (inducing a one parameter- β - family of solutions), then reducingto IIA and T-dualizing back we had our final generated background.One may wonder what is the algorithm when, after a number of T-dualities, we end witha background solving the Type IIB supergravity equations of motion. In this case, a way ofgenerating a one-parameter family of solutions may be S-dualizing. Indeed, given a, b, c, d realnumbers satisfying ad − bc = 1, we can start with a IIB solution having axion χ , dilaton φ , RRand NS thre forms F = dC and H = dB and five form F = dC + F ∧ B and by S-duality,generate a new solutions dependent on the three independent real parameters a, b, c . Indeed,the five form is left invariant and the same happens for the Einstein frame metric, while F [ new ] = bH + aF , H [ new ] = dH + cF ,e φ [ new ] = (cid:16) ( cχ + d ) + c e − φ (cid:17) e φ , χ [ new ] = ( aχ + b )( cχ + d ) + ace − φ (cid:16) ( cχ + d ) + c e − φ (cid:17) (D.1)Let us focus our attention in the particular set of values a = d = 0 , cb = −
1, this is thetransformation that interchanges the three forms and inverts the value of the dilaton (if theaxion is initially zero, as we will assume).One can then think about applying this transformation expecting to generate new inter-esting solutions. As an example, suppose that we start with solutions describing D6 braneswrapping a three-cycle inside the deformed conifold. Those solutions are dual to a (UV com-pleted version of) N=1 SYM [29]. A particularly interesting solution is given in section 3.2 of27he paper [30]. The background consist of metric, dilaton and RR two-form field, ds IIA,st = e f [ − dt + dx + dx + dx + dM ] , e φ = e f +2 φ , F = dA (D.2)where all the details (the manifold M , the functional form of f, φ and the one form A aregiven in eqs(57)-(59) of [30]). We can the perform three T-dualities in x , , leading to ds IIB,st = e f [ − dt + dM ] + e − f dx , , , e φ = e φ ,F = dA ∧ dx ∧ dx ∧ dx (D.3)that is we have generated D3 brane charge and the dilaton is just a constant. We now movethis to Einstein frame, that leaves us-up to a constant-with the same metric, perform the S-duality metioned above and T-dualize back. This brings us to the starting point background(D.2) . Something similar occurs if we start with a solution describing D5 branes wrapping athree-cycle inside a G2 holonomy manifold [31], a configuration dual to N=1 Yang-Mills Chern-Simons in 2+1 dimensions (with its respective UV completion). Notice that this operation isnot the one proposed in [20], that is the reason why these authors were able to generate aninteresting solution starting from [31].In both these cases described above, we are generating charge of D3 brane, this is invariantunder the S-duality, hence it is expected that the whole operation brings us back to the initialconfiguration. A more interesting example is to start from the configuration of D4 braneswrapping a circle with SUSY breaking boundary conditions [17] that we discussed before, seeSection 2.1.1.In this case we will apply three T-dualities to a IIA configuration. This will generate chargeof D1 brane that the S-duality will interchange with F1 charge, after T-dualizing back, we willgenerate a new background (the SO (1 ,
3) isometry will be spoiled, so we may want to startwith the high Temperature dual, hence having a non-extremal factor h ( ρ ) in front of dt andthe function f ( ρ ) = 1 in front of dx , but let us keep things general). Let us see some details.After the T-dualities, the configuration reads ds IIB,st = H − / ( − hdt + f dx ) + H / ( dρ h + ρ d Ω + dx , , ) ,e φ = g s H, F = ∂ ρ Adt ∧ dx ∧ dρ (D.4)now, we need to move this to Einstein frame, multiplying the metric by e − φ , perform theS-duality that will generate H and a dilaton e − φ = g s H . Then, T-dualize back in x , , . Thefinal configurations is, ds IIA,st = c / g s H ( − hdt + f dx ) + dρ h + ρ d Ω + dx , , ,H = c∂ ρ Adt ∧ dx ∧ dρ, c e − φ = g s H. (D.5) One may perform an S-duality with parameters a = 0 , bc = − d being free. This will generate abackground in IIB with constant axion-dilaton. The final IIA configuration is the same as the initial one.
28s a final remark; had we chosen to T-dualize only in the x direction (in the supergravityapproximation, this can be done in the high Temperature phase only), we would have gen-erated D3 branes, hence after the S-duality and T-duality, we would be back to the initialconfiguration. E Appendix: U-duality for the wrapped D5 branes BlackHole
We will describe in detail the action of the solution generating technique proposed in [6],when applied to the background of eq.(3.30). These techniques have been applied in a similarcontext in [39]. Let us consider things in the string frame. ds s = e φ h − h ( ρ ) dt + dx + dx + dx i + ds ,s ds = e φ h e k s ( ρ ) dρ + e k ω + cos θdϕ ) + e q ( dθ + sin θdϕ ) + e g d ˜ θ + sin ˜ θd ˜ ϕ ) i ,F (3) = N c " − ˜ ω ∧ ˜ ω + sin θdθ ∧ dϕ ∧ (˜ ω + cos θdϕ ) , (E.1)where h ( ρ ) , s ( ρ ) are the non-extremality functions and we will choose h ( ρ ) = s ( ρ ) as we didin section 3. We can proceed to rotate it. We will follow the procedure explained in [6]. So,let us start by writing the effect of the first T-duality in the x direction (all the expressionsbelow are in string frame) ds IIA = e f h − hdt + dx + dx i + e − f dx + ds ,e φ A = e φ − f , F = F ∧ dx (E.2)The function f = φ will be kept to avoid confusion with the transformed dilatons. Now, weperform the T-duality in x ds BIIA = e f h − hdt + dx i + e − f ( dx + dx ) + ds ,e φ B = e φ − f , F = F ∧ dx ∧ dx (1 + ∗ ) (E.3)T-dualizing in x , we get ds IIA = e f h − hdt i + e − f ( dx + dx + dx ) + ds ,e φ A = e φ − f ,F = F ∧ dx ∧ dx ∧ dx → F = e f √ h ∗ F ∧ dt (E.4)29otice that ∗ F = N c √ h h − e q − g sin θdθ ∧ dϕ + e g − q θd ˜ θ ∧ d ˜ ϕ i ∧ dρ. (E.5)The factor of √ h in the F of eq.(E.4) is present to cancel the factor of √ h in the denominatorof eq.(E.5). Now, we lift this to M-theory; ds = e / φ − f dx + e f − / φ h − he f dt + e − f ( dx + dx + dx ) + ds i ,G = √ he f ∗ F ∧ dt. (E.6)We boost in the t − x directions according to, dt → cosh βdt − sinh βdx , dx → − sinh βdt + cosh βdx (E.7)and now we rewrite this boosted metric as, ds = e f − / φ h e − f ( dx + dx + dx ) + ds i + Adt + Bdx + Cdtdx ,G = √ he f ∗ F h cosh βdt − sinh βdx i (E.8)where, A = e f − / φ [sinh βe φ − f − h cosh β ] , B = e f − / φ [cosh βe φ − f − h sinh β ] ,C = − β sinh βe f − / φ [ e φ − f − h ] . (E.9)Now, we will reduce this to IIA, before doing so and in order to reduce to IIA, it is useful torewrite eq.(E.8) as, ds = B − / h g tt dt + B / e f − / φ ( e − f ( dx + dx + dx ) + ds ) i + B ( dx + a t dt ) ,G = √ he f ∗ F h (cosh β + a t sinh β ) dt − sinh β ( dx + a t dt ) i (E.10)where we have defined a t = C B , g tt = 4 AB − C √ B , e / φ A = B. (E.11)Now, we reduce to IIA, obtaining in string frame, ds IIA = g tt dt + √ Be − / φ ( dx + dx + dx ) + √ Be f − / φ ds ,e φ A = B / ,F = √ he f ∗ F ∧ h (cosh β + a t sinh β ) dt i ,H = − sinh β √ he f ∗ F ,F = a t dρ ∧ dt (E.12)30ow, we proceed to do the T-dualities back; T-dualizing in the x direction we have ds IIB = g tt dt + √ Be − / φ ( dx + dx ) + e / φ √ B dx + √ Be f − / φ ds ,e φ B = Be / φ ,F = √ he f ∗ F ∧ h (cosh β + a t sinh β ) dt ∧ dx i (1 + ∗ ) ,H = − sinh β √ he f ∗ F ,F = a t dρ ∧ dt ∧ dx (E.13)now, we T-dualize in x ds IIA = g tt dt + √ Be − / φ ( dx ) + e / φ √ B ( dx + dx ) + √ Be f − / φ ds ,e φ A = √ Be / φ ,F = √ he f ∗ F ∧ h (cosh β + a t sinh β ) dt i ∧ dx ∧ dx ,H = − sinh β √ he f ∗ F ,F = a t dρ ∧ dt ∧ dx ∧ dx (E.14)finally, we T-dualize in x ds IIB = g tt dt + e / φ √ B ( dx + dx + dx ) + √ Be f − / φ ds ,e φ B = e φ ,F = √ he f ∗ F ∧ h (cosh β + a t sinh β ) dt i ∧ dx ∧ dx ∧ dx F = ∗ F ,H = − sinh β √ he f ∗ F ,F = a t dρ ∧ dt ∧ dx ∧ dx ∧ dx (1 + ∗ ) . (E.15)After using f = φ and the definitions for A, B, C, a t this encodes the result of eq.(3.35). F Appendix: The equations of motion
In this appendix we will quote the equations of motion that we are numerically solving. Ourgoal is to find a black hole of the metrics described in the main text. In [23] the authorsstudied a very general Ansatz for non-extremal deformations of
N S S .Their Ansatz can be adapted to our case. It reads –in Einstein frame, ds = − Y dt + Y d x n d x n + Y dρ + Y ( e + e ) + Y (cid:0) ( w ) + ( w ) (cid:1) + Y ( w + A ) (F.1)31nserting this ansatz in the supergravity action we get, L = X i,j G ij ( Y ) Y i Y j − U ( Y ) = T − U (F.2)Using a parametrization to make G ij diagonal and choosing the appropiate gauge to makecontact with our ansatz we have, T = e g + q − x +Φ) (cid:18) (cid:0) g + q − (cid:1) + 2( g q − k x ) + ( g + q + Φ )( k − x + 2Φ ) (cid:19) ,U = 1256 e − g + q − Φ) (cid:16) − e g + q + k ) ( e g + 4 e q ) + ( e g + 16 e q )(1 + e k ) (cid:17) . (F.3)and, Y = e Φ / e − x , Y = e Φ / , Y = e Φ / e x e k ,Y = e Φ / e g , Y = e Φ / e q . Y = Φ (F.4)¿From (F.2) we get the second order equations to solve,2 e − g +8 x + 18 e − q +8 x − ( g + q − x + Φ ) − Φ = 0 − x ( g + q − x + Φ ) − x = 0 − e k +8 x (2 e − g + 18 e − q ) + e x (2 e − g + 18 e − q ) + 2 k ( g + q − x + Φ ) + k = 0 (F.5)2 e x ( e k − g − e k − g ) + 1) + g ( g + q − x + Φ ) + g = 0 e x ( − e k − q + e k − q − e − q ) − q ( g + q − x + Φ ) − q = 0and a first order constraint which is a consequence of reparametrization invariance, e x (cid:16)
116 ( e − q + 16 e − g )(1 + e k ) − e k ( e − q + 4 e − g ) (cid:17) + 12 ( g + q − )+2( g q − k x ) + ( g + q + Φ ) ( k − x + 2Φ ) = 0 (F.6)The second equation in (F.5) can be integrated to yield a first order equation and (F.5)becomes 2 e − g +8 x + 18 e − q +8 x − ( g + q − x + Φ ) − Φ = 0 x − αe − g + q − x +Φ) = 0 − e k +8 x (2 e − g + 18 e − q ) + e x (2 e − g + 18 e − q ) + 2 k ( g + q − x + Φ ) + k = 0 (F.7) e x ( e k − g − e k − g ) + 1) + g ( g + q − x + Φ ) + g = 0 e x ( − e k − q + e k − q − e − q ) − q ( g + q − x + Φ ) − q = 032here α is a non-extremality parameter.One can verify that the transformation r → e d r Φ → Φ + d x → x + d g → g q → q k → k (F.8)where d is a constant, leaves (F.5) and ( F.6) invariant and thus, is a symmetry of the equationsof motion and constraint. Another, obvious, symmetry of (F.5) and ( F.6) isΦ → Φ +
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