JHEP04(2001)033 × ) M + N b April 27, 2001 , fractional D3-branes at Accepted: M = 1 supersymmetric SU( regular and N March 20, 2001, N and Christopher P. Herzog ab [email protected] , Received: Igor R. Klebanov ∗ a Duality in Gauge Field Theories, D-branes, Black Holes in String The world volume theory on [email protected] [email protected] [email protected] ) gauge theory. In previous work the extremal Type IIB supergravity dual of HYPER VERSION N Department of Physics, Theand Ohio State University, Columbus,Blackett OH Laboratory, Imperial 43210, College, USA LondonE-mail: SW7 2BZ, U.K. Michigan Center for TheoreticalAnn Physics, Arbor, The MI University 48109, ofE-mail: USA Michigan Joseph Henry Laboratories, Princeton University Princeton, New Jersey 08544,E-mail: USA Institute for Theoretical Physics, UniversitySanta of Barbara, California California 93106-4030 [email protected] b a SU( this theory at zero temperaturea was deformation constructed. of Regularity the ofTo conifold: the study solution this the requires is non-zerobe a temperature considered, reflection and generalizations in of non-extremal the thebe high solutions restored. chiral temperature have symmetry phase Such to the a breaking. chiral solutionconstruct symmetry is an is expected expected ansatz to to have necessarythe a to simplest regular Schwarzschild possible study horizon. solution such has We non-extremalorder a equations solutions singular in horizon. and the show We radial derive that variable the whose system of solutions second mayKeywords: have regular horizons. Theory, AdS-CFT Correspondance. the conifold singularity is a non-conformal Abstract: Arkady A. Tseytlin Leopoldo Pando Zayas Alexander Buchel, Non-extremal gravity duals forD3-branes fractional on the conifold JHEP04(2001)033 15 =1 (1.1) .In[2] N r =0 w 3 F ∧ 3 = 0, here the 3-form field strengths are H 1 5 = ) gauge theory. dF 5 M dF + N SU( × ) N 3.1 Effective 1-d3.2 action for radial The evolution superpotential3.3 and the extremal The KT full solution system of 2-nd order equationscases5.1 Standard regular5.2 non-extremal D3-brane solution Special singular non-extremal D3-brane solution 6 5 7 12 13 11 Also at Lebedev Physics Institute, Moscow. ∗ does not vanish. Inthis fact, behavior the 5-form was flux attributedsupersymmetric increases to SU( without a bound cascade for large of Seiberg dualities in the dual 1. Introduction In this paper wedescribes study regular and non-extremal fractional generalizations D3-branesKT of at solution the the is apex KT of crucially solutionIIB the dependent conifold. [1], supergravity. on The which the These extremal presencewhile terms of cause in Chern-Simons the regular terms RR D3-brane in 5-from solutions type flux to vary radially. Indeed, 4. A singular non-BPS generalization5. of the KT General non-extremal solution pure D3-solution and its regular and singular 6. Discussion 8 A. Non-BPS solution with non-constant dilaton and 14 Contents 1. Introduction2. Non-extremal generalization of the3. KT ansatz Basic equations 3 1 5 turned on in such a way that the right-hand side of the equation JHEP04(2001)033 ) M + N =4SYM The sym- 1 SU( N . c × T .Inthatcase,a ) ∞ N for this particular = c c T T = 0 theory exhibits con- T , there should be non-extremal c T 2 is the relevant number of colors. This factor rule out the possibility that N ). The mechanism that removes the naked singularity where ρ a priori 2 N by the deformation of the conifold [2]. chiral symmetry, which may be approximated by U(1) for 2 M Z 2 Z [6]. One expects that turning on finite temperature in the field the- c T [7, 8]. The difference is that in our case the , in agreement with color confinement. It is interesting, therefore, to study 0 ,isbrokento 5 N M One implication is that, at temperatures below Further examples of supergravity backgrounds with varying flux were constructed In [6] a different mechanism for resolving this naked singularity was proposed. Strictly speaking, we cannot = 1 gauge theory. The proposal of [6] is that the description of the phase with AdS 1 theory, which is notin confining, is described at a finite temperature by a black hole indicates that the colorthen degrees the of entropy could freedom only appear areorder through not string confined. loop effects If which would there make it is of no horizon, generalizations of the KSsolution. solution which The are absence freeby of of the a horizons, resulting horizon just area is likeof law a the freedom. for manifestation extremal Wilson of Once loops confinement, thewith or, as it horizon typically alternatively, evidenced appears, by scales the counting as Bekenstein-Hawking of entropy degrees associated finement and chiral symmetryshould breaking appear [2]. only forunusual So, some and, the to finite regular our Hawking Schwarzschild knowledge,of temperature. horizon unstudied view. phenomenon This from would the be supergravity point a rather this theory as awhere function the chiral of symmetry the is temperature, restored. and to identify the phase transition N restored symmetry involves a regulartotically Schwarzschild horizon KT appearing geometry. in This the proposal asymp- is analogous to the fact that the metry restoration is part of the deconfinement transition at large regular black hole inway KT to geometry decide does thehorizon not does issue exist. shield is the While to singularity. this find possibility seems the strange, actual the black only hole solution where a regular Schwarzschild ory, which translates intorestoration non-extremality of on the the chiral supergravity symmetry side above [7, some 8], critical leads temperature to in [2]–[5]. In particular, itgularity is present important in to the understand KT thedeformed. solution. resolution of As The the a proposal naked result of sin- singular of [2] and this is without deformation that a the the horizon extremal conifoldsolution in KS becomes in the solution the IR, is UV while perfectly (for it non- large asymptotically approaches the KT is related to the breaking of the chiral symmetry in the dual SU( gauge theory. The It was suggested thata a regular non-extremal Schwarzschild generalization horizontheory of interpretation “cloaking” the of the KT this naked wouldtemperature solution be singularity. may the have restoration The of dual chiral symmetry field at a finite JHEP04(2001)033 (2.1) (2.2) (2.3) which x, y, z, w ∞ =0)and = 00 u G , 2 5 ds 6= 0. The vanishing , w 2 ) 2 6 ∗ e r ds ( ≡ z 1 2 ) − 2 e 2 φ e , one should not impose self- ’s involves 4 functions P + )+ 2 i 2 S 2 θ e dX i + 1 dX 2 φ , x e 2 2 ) e 5 + 3 1 + 2 θ 2 0 e dM ( ( y w 2 dX 2 e x e 6 + − + e 2 ( 2 ψ z e 2 du w e y 8 10 − = e e on another branch of the solution. This type of singular E u = = . Our ansatz will be more general than that of [6]. It turns 2 10 in [6, eq. (2.60)] is incorrect because ∆ ∞ 1 , 2 6 2 ∗ 1 ) ds r 5 = ds T = r dM r ( -rotations and the interchange of the two ψ ) is an artefact of the coordinate choice. We find a good radial coordinate ∗ r ( 2 Nevertheless, the scenario for chiral symmetry restoration proposed in [6] is very A general ansatz for a 10-d Einstein-frame metric consistent with the U(1) sym- In order to address these questions from a dual supergravity point of view, we Thus, the solution found in [6] does not have a regular Schwarzschild horizon and show that the solution derived in [6] has a singular horizon at where of a radial coordinate u of ∆ the singularity are coincident,rameter. independent of We also the show choice(wrapped that, of D5-branes), in the and the non-extremality leave limit pa- onlynot where reduce the we to regular remove the D3-branes, the standard non-extremal thestandard fractional 3-brane solution D3-branes non-extremal metric. of version Instead, [6] it of does horizon. reduces a to a D3-brane non- solution whose metric has aappealing. singular This motivatesand us to to begin introduceSchwarzschild search a horizons. for more solutions general that U(1) are symmetric asymptotically ansatz KT but possess regular duality on the 3-formsadopt away a from more extremality. generalconstant This dilaton. ansatz in for turn the implies metric that one than is inmetry of to [6] and also allow for a non- out that in orderD3-brane to in the look limit for of solutions zero which fractional reduce brane charge to the standard non-extremal 2. Non-extremal generalization of theWe KT start ansatz with animpose ansatz the for requirement that thethe the non-extremal U(1) background KT fiber has background. a of U(1) Just symmetry associated as with in [1] we need to study non-extremalproblem generalizations was of first themethods. addressed KT for We and rederive the KS thisthe KT backgrounds. solution horizon as background analytically This and in show [6] that partly the with identification of numerical corresponds to horizon is deemed unacceptable in studies of blackshielding hole the metrics. naked singularity ofing the extremal temperature. KT metric for Instead, sufficiently high the Hawk- horizon (defined as the locus where JHEP04(2001)033 = = x P 0or 4 (2.8) (2.5) (2.4) (2.6) (2.9) (2.7) : − . is the (2.10) z i P 4 → )drives . w − u dφ ∗ 2 ( i ρ θ x ,e , dρ 4 ] ) 2 5 ρ is removed by sin . . ( ds 1 6 a/ρ ∗ 2 , 2 1 ∞ 2 φ − ρ ] √ ρ e 2 − 2 φ = ∧ + = e =g ρ< 2 2 i ρ 2 θ + φ e ≤ =1 dρ 2 du ) 2 θ ∗ ∧ g x ρ e 1 ( ,e 1 +2 φ i = + y e − 1 x 10 dθ 2 φ 8 ∧ e )[g 6 relative to the 2-spheres; it does ,ρ − 1 1 ρ θ 1 w √ 6= 0 analogs in the case when the ( , + e 2 , 1 + 1 / y ) 2 θ ∧ w = 1 T e , and we make similar rescaling of ,e 2 , e h i φ ψ T ) x θ au, e e e 8 2 = 2 + 1 ) φ − √ ∧ = ]+ e u 2 ψ e i 4 ( e 2 2 ,e = x θ ) ∧ = ) K e ρ 2 f dX 2 are the 3 longitudinal 3-brane directions. ( θ i e i ,ρ − κ = [ dφ by 1 6 ∗ 6 2 X 2 =0. dX − φ ), the transverse 6-d space is the standard conifold F ρ ρ ρ f θ e 1 u w + = 0 (for their . The limit of the standard conifold is φ ,g + − ∧ 2 e 0 (4 ψ w 2 w 1 / + θ ∧ , x +cos e dρ dX 1 =1 = ( ) ) )andviceversa. + θ 1 =1 y ρ e ρ w Our aim will be to understand how switching on the non- thus corresponds to large distances (where we shall assume -form fields is dictated by symmetries and thus is exactly ∗F x ∧ ( e ( 4 dφ p 1 )( . 10 g ρ ψ u 1 + − x u = − )[ θ ( →∞ e + κ ρ = y Pe f F squashes the U(1) fiber of ( ) changes the extremal KT solution. ρ e y 2 = 4 / w = = = in [1] is related to 1 au e )= 2 6 = as +cos − 3 2 5 ρ , T .Small ( ,ρ = ds h F F B 1 1 , κ x ,ρ = 0) conifold solution has dψ x 1 4 = ( 4 − T → 3 1 z which corresponds to w E 4 is the euclidean time and = =0and q/ρ 2 10 − = e 5 0 w ds . ψ X e M = h, g, v The extremal D3-brane on the conifold and the more general fractional D3- Our ansatz for the The function This metric can be brought into a more familiar D3-brane form KT The function identification of the angle h →−∞ 2 P =1+˜ 2 2 ∗ 1 √ the same as in the extremal KT case [1]: When with the redefinitions Z h that y 6-d space is the generalized conifold see [9]). Adding a non-constant brane KT solution have This space has regular curvature, and a bolt singularity at Here extremality ( and with not violate thegeneralized U(1) conifold symmetry. of [9] A Ricci-flat 6-d space with non-trivial the non-extremality. Forstandard example, ( the non-extremal version of a D3-brane on a JHEP04(2001)033 5 u m . L M i 0. 2 =0, (3.1) (3.2) ) (2.11) → w . Pf P )] w +2 12 , implies Q − 3 ( e z F 8 = 0 limit leads to e − ∧ w 3 P + 2 2 H − e P = w -dependent factors): (6 5 y w 8 +4 y e dF +4 + , i.e. that the 3-forms are self z = y 2 0 . 5 . Thus, following [1], ) can no longer be held constant. In w F = 0 the regular D3-brane solution Φ+4 x Φ+4 u 5 e ( ∗ w w − d Pe +2 2 Pf of the effective lagrangian is essentially 0 − 2 for z 0 m 2 f = +2 (up to angle-dependent factors), and com- 5 L w 0 − 4 z produces exactly the same 1-d gravitational Q f 2 2 − = Φ = 0 are the KT solution [1] and its non- 0 − y 5 x 4 y w S 3 − )= 10 z e u − ( 1 2 0 Φ+4 108 → K analytic form and clarifying its geometry. Its horizon y − or by ] e = [5 1 , 1 ··· G du +2 T √ 2 + 0 Z explicit version will not change if we replace the flat transverse space 2 is not squashed. If the fractional branes are present, then the Φ h 1 1 , 27 Φ) 1 1 8 ∂ T ( 2 1 → du − and Φ. We shall then discuss its solutions generalizing the work in [1] [ Z = 0 is a consistent fixed point of the equations of motion. Replacing GR G 1 27 w √ √ x non-extremal x 10 10 d d →− Demanding that the non-extremal solution have the correct For the metric (2.1) It is easy to see that the matter part In what follows, we shall derive the corresponding system of type IIB supergrav- The simplest special fixed-point solution of our system turns out to have with the conifold. x, y, z, w, f Z Z 6 the next section we derive the resulting system of second-order differential equations. BPS generalization considered in [6].in We will section discuss 4, the finding latter solution its in some detail only U(1) symmetric solutions with turns out to be singular.non-extremal D3-brane A solution related in problem the is limit that of it no does fractional notthe brane reduce necessity charge, to of the relaxing regular the condition Note that and its lagrangian. That means, in particular, that dual. If the 3-forms are not self-dual, then Φ and the same as it was in the extremal case [1] (apart from does not depend on the non-extremality function in (2.3) by the standard 3.1 Effective 1-d action forAs radial in evolution [1], the mosttions efficient of way motion to is derive toradial the follow evolution. system [10, of 11] type and to IIB start supergravity with equa- the 1-d effective action for the 3. Basic equations As in [1], the Bianchi identity for the 5-form, puting the scalar curvature we find R ity equations of motion describing— radial evolution of the sixto unknown the functions non-extremal of case. i.e. the case when JHEP04(2001)033 W, j term (3.5) (3.8) (3.4) (3.7) (3.9) (3.3) (3.6) (3.10) 2 W∂ . a i 0 ∂ − , ij W = 2 2 g − 2 ) i 2 2 0 − ) ) − x w , or the angular w ) 6 Pf = , − uu +4 2 e W V y 0 l G +2 f ∂ , − . w jl Q Φ+4 ) 4 G w . ( g )as 4 − z 0 √ φ e 8 y Pe + ( 4 ( Pf e y j − > 0 8 1 4 + z , W e =0. φ 0 0 +2 + 5 3 f i )( ( ) Φ+4 V 2 Q w − ( − W P 4 z + e 0 k , Pf 4 − w 4 1 ∂ y e w =const T 4 h +4 8 1 ik − y − 5 +2 g =0 z 2 − 0 +4 Q + z ) − ( Φ W i Φ+4 z 0 w 2 k 8 1 4 6 − 6 φ i ∂ does not appear in Φ+4 e e − ( ) e − 8 ik 1 e 1 4 x w ij au , a 4 1 g 6 2 2 g is simply linear 0 and thus is a “modulus” — it has no potential i − − = ) w + +2 e u ) m 5 i )+ 0 w w )= L 6 4 w φ Pf φ − +2 e − ( 12 2 e 0 w − (3 V 4 +2 z y e . ,x e 4 2 x +2 − e Q − (3 + ( j − 4 1 0 y w z 2 w 4 4 =0 0 4 2 φ 2 0 e e i e = 0 00 Φ − x 5 1 4 1 x φ (3 e 3 8 1 ) y W 4 + (6 φ V, + − − e 0 y ( 0 8 2 y 1 4 2 0 ij z − 0 e h g h h y x ) can be expressed in terms of a function 2 2 T − 3 φ = ( − − − +5 = =5 = ) is the kinetic term metric, then V V = L φ T V ( − L ij T g = Since the non-extremality function L 3.2 The superpotential and theThe extremal crucial observation KT made in solution [1] isspecial that the structure lagrangian — (3.4), (3.5) it hasthe admits a following remarkable a way superpotential. Indeed, it can be be represented in appearing in the “zero-energy”solution constraint) for which a spoils non-constant the BPS nature of the KT supplemented with the “zero-energy” constraint As a result,equations, there satisfying exists a special BPS solution of the corresponding 2-nd order where the last term isin general, a total derivative and may be dropped. Let us recall that, if, and thus also the zero-energy constraint. As follows from (3.7), in the present case [1] where part of the metric, it is absent in From (3.1) and (3.2) weevant get overall the numerical following factor) 1-d effective lagrangian (ignoring an irrel- (cf. (3.1)). Thus its dependence on This is just what is expected of an extra kinetic energy (i.e. the JHEP04(2001)033 (3.12) (3.16) (3.15) (3.14) (3.11) (3.13) 6= 0, i.e. 0 x , leads to frac- . , w 0 )=0 w 3 Pf 6 )=0 u. − ρ, e ln +2 Pf u ln − 2 Q 4 2 2 w +2 ρ P 4 = P 2 e Q ( 0 − ( y ρ, + z 4 u 4 , e 4 ln e ρ 5 3  1 4 1 4 2 P -equation in (3.12) we then find the 2 =0 − P + , ,K  0 + w 0 0 2 1 Φ + 2 1) u, f P 0 − ln = + 7 Φ=0 Pf u 4 1 P , ,z ,w (ln − +2 Pf coordinate, see [9, eqs. (4.20), (4.21)]. While the gen- =0ofthe u 0 =0 =0 , ρ 2 Q f ,f 0 2 w )=0 +2 w in (3.6) is non-zero. In order to do that we need to P  x used in [1], w = u 1 6 =0 Q a +4 4 ρ − y −  e u )=0locus. = 0 u Φ+4 4 ( =1+ +2 K ρ K h Pe w ,w 4 =1+ = u, f = e + y h in (3.13) implies self-duality of the complex 3-form field, i.e. 0 z 4 (3 4 e f =0 =4 =1+ y ln 2. − f 4 z y e x e 2 P 4 4 5 1 − − e e + does not depend on Φ. The corresponding system (3.9) of 1-st order + 0 . 0 3 f y W = ?F 1 Φ f e A more general extremal solution of (3.11)–(3.13) with non-zero Choosing the special solution For its explicit form in terms of the = 3 3 the non-extremality parameter start with the original 2-nd order system following from (3.4), (3.5) or (3.7), i.e. the KT solution [1] i.e. tional D3-branes on the generalized conifold (2.6) solution of [9]. 3.3 The full system of 2-ndWe would order like equations to generalize the solution (3.15) to the non-BPS case when Note that equations is then [1]: where eralized conifold has, inD3-branes contrast makes to the the metric standardcoinciding conifold, singular: with regular pure the curvature, singularity. D3-branes the Thenaked on back fractional singularity the reaction behind D3-brane of generalized solution the conifold is have similar a to horizon the KT one: it has a In terms of the radial variable The equation for H JHEP04(2001)033 =0, (4.4) (4.2) (4.3) (4.1) (3.17) (3.18) (3.21) (3.22) (3.19) (3.20) w +8 = 0, i.e. y V . , , . , , 2 + 2Φ+8 a e — and we will 2 T =0 =0 =0 , P )=0 )=0 z 00 00 =3 8 2 2 0. From (3.18), if Φ f,z e 2 − P P ) ) − 2 2 w w − 0 0 )=0 )+Φ ) f f and Φ Pf Pf +8 +8 a> 0. w w w y y 4 Pf 12 12 will also be forced to run. − , , , +2 +2 = − − y e 4 e 2Φ+8 2Φ+8 w 0 P> +2 Q Q − e e ( x ( z − − =0 z Q P 8 + − ( w w y,w,z,f )=0 )=0 00 e 2 2 z 2 2 Φ+4 0 w w 0 4 − Φ 8 1 − − should be subject to the first-order − e f f 0 0and 12 12 e e e ( ( ) ( 4 1 z 0 − − + 4 1 w y w (6 e e f 4 8 2 4 y + e 8 w − − a> − P − − 0 4 y e y 4 4 2 w − 8 12 0 w w 8 y − − 2 2 4 z z , the function Φ +4 − − − − y − is not squashed. We will discuss this class of f 8 1 e e z 00 00 ( 1 +4 y Φ+4 Φ+4 , ,z y w (6 z − 1 8 − − y Φ+4 2 e 10 8 e e 0 T 10 − e 6 Φ+4 =0 w e 4 + − dynamics constitute a novel phenomenon specific to e ( 5 − z w 00 4 1 − 8 00 w − Φ e +4 00 D3-brane case, 2 w y 2 y 0 ) 5 )+ 5 z w 2 Φ+4 are subject to (3.13), i.e. Pf 12 − − z =0sothat Pe e 2 , then (3.21) implies that 0 +2 w y w (3.6) plus a coupled system for + − 5 0 Q fractional and +4 x ( w f y 2 f − − e Φ+4 00 z (6 (3.13) are still satisfied, while (3.11) and (3.12) are replaced by their 2-nd 4 y Pe 8 z e − − = and 0 In general, if we relax the 1-st order conditions (3.13) on f f the non-extremal Hence, we need theThese nontrivial more dilaton general and metric (2.1) in order to have a regular horizon. we relax the first order constraint on be forced to relaxaccording them to in (3.19), order driven to by have a the regular non-extremality horizon — the dilaton will run The integration constants are subject to the zero-energy constraint This system has special propertiesparticular, reflecting there the is structure a offor the subclass lagrangian of (3.7). simple In solutionsorder for counterparts. which the Indeed, 1-st it order is equations easy to see that if we set i.e. solutions first in the next section. so that (3.20) and (3.21)(3.18), are (3.19) satisfied automatically, and the (3.22) remaining become equations (3.17), Assuming that 4. A singular non-BPS generalization of the KT solution free equation for equation in (3.13).particular, In to this keep case the 3-forms are self-dual. Then it is consistent, in JHEP04(2001)033 . u (4.5) (4.7) (4.6) (4.8) which is K in (2.4) will is obviously b x 4 − ∼ z subject to (3.12), 4 a − . e singularity coincides 2 will keep growing at , one is then to plug a y,w = z w 4 h − e )=3 and . a. w z 3 5 y 12 =0and − r a e and bu , , i.e. here the = − . x f 4 w 2 − bu e − e b 0) asymptotic conditions we then have is not squashed, and the eqs. (4.2)–(4.5) (6 ,b ln tanh 2 y 9 y 1 8 sinh 4 , → 8 4 P e 1 and has the KT behavior (3.14) at small e u = T − u − + y 2 ∗ 4 0 2 0 we will see that we recover the non-BPS gener- f b e Φ √ 8 1 = − f − P> at large 2 = 0 0 w stays positive (does not change sign) so does y 4 5 f − due to the vanishing of 2 0 (cf. (2.11), (4.1)). That means that y constant ∞ z 5 4 (short distance) behaviour is an improvement compared to the ex- = − . e u u =Φ=0.Inthiscase,the = 0 limit this solution does not become the standard (regular) non-extremal w approaches a Integrating (4.2) and using (4.5) we find We begin an analysis of the non-BPS solutions by discussing the special case This is a rather peculiar situation, different from the naked singularity of the The large The BPS solution of (4.2)–(4.5) corresponds to P f Other known cases of solutions where a horizon coincides with a curvature singularity include, 4 Notice that the matterbe and gravity obtained parts just are by nowmetric looking decoupled: in for eqs. the Ricci-flat (4.2)–(4.5) class uncharged would black (2.1). 3-brane Finding solutions with first a the functions crucial for this difference. Integrating (4.1) shows that Assuming the required long-distance ( small distances, and does not go to zero. The total function D3-brane solution. Similar singular non-BPSconstructed solutions in with appendix non-constant dilaton A. are simplify substantially. When alization of the KTthe solution first discussed in [6]. In section 5.2 we will see that in i.e. a derivative of and vanish only at them back into (4.1) to determine the functions leading to the generalized conifold space (2.6), (2.7). when KT solution. The non-vanishing of the non-extremality parameter with the horizon for example, dilatonic BPS Dp-branes [12] as well as D3-branes on generalized conifolds [9]. tremal KT case: since JHEP04(2001)033 . u y 6 − 6=0. z (4.9) . 6 (4.13) (4.12) (4.10) (4.11) ? e x 2 . P →∞ )], we find , ∞ (0) = 0. = u u h ( = R ··· u Pf + , bu = 0 and grows as +2  4 ) u − Q bu e [ asymptotics, it is clear 4 2 − P u e . du ( + 2 R is 1 at ∗ bu , ··· Li f z = 4 + − : even though the Ricci scalar and shifts the singularity of the − = z ) e 4 bu ∞ to the same point − sinh 4 4 bu ∞ , may be adjusted so that the solution has is singular in the limit e 4 − u − = e au = C au e R 8 2 4 b u u − ,f 4 − − P e ( e 2 b a − ··· 10 3 4 Li in (4.1) u  ∗ )) = + ) was found numerically to vanish for z 2 b ? u K ( )= 8 bu x P 8 ( x u + ( ( − f + . The constant e f x 0 limit of (4.7)–(4.9), thus removing the fractional . Moreover, from (2.1) and the differential equations b C 4 1 u ∗ 32 ≡ ≡ = → / + K ) 2 x z b x 4 →∞ P P ( + 2 − ) vanishes, then from these large .In[6], u f e π x C A 5 log 2 6 . − 4 = : 4 1 ∗ 243 C − u z 4 z = + Pf = − asymptotics, it is clear that e a (0) bu u refer to the notation in [6]. z ) that determines the position of the event horizon is solved with the +2 4 − x x − 6 ( = 0 we get from (4.9) the same expression as in the extremal case Q e = f always grows never reaching zero. If we define the horizon as the locus =exp(2 = u y and = 0), i.e. the KT behaviour (3.15) where p z ∗ 00 4 f − can be expressed in terms of polylogarithms ≡ G K ,b e ) z Near We conclude that the non-BPS solution (4.7)–(4.9) does not have a horizon The exact analytical solution presented here realises the non-BPS KT back- Explicitly, from the equation for In fact, if we take the τ Here Note that ( =0 6 5 1 a vanishes, components of the curvature tensor blow up. The lesson is that the presence Thus where Given (4.10), (4.11) we seeto that the this wrong statement is conclusion incorrect.KT of geometry the This proposed numerical nonsingular error in horizon led [6]. of the black hole solution in the ( that we have a horizon as From the large shielding the naked singularity of theof extremal non-extremality KT here solution; instead, creates the a introduction horizon at extremal KT background from a finite value of branes, we still have a singular horizon at or does not have an asymptotically flat region; the latter possibility corresponds to that provided we set ground discussed in [6].4 In particular, the equation (2.53) in [6] for the warp factor where increases. For large (4.1), (4.2) and (4.5), we find that the Ricci scalar in this space-time is identification JHEP04(2001)033 the (5.4) (5.2) (5.1) (5.3) (3.18) and , y Q, au 4 1 4 e in (2.5) approach ≡ = . x 5 h, g 4 S To match the standard , = ,q 7 5 z Then we are left with (3.6) 8 M =0 to the ordinary non-extremal ,e e 8 2 z = 0 case, i.e. to singularity of =0. ) 8 q k e and f P . not 2 + 1 0) and that , + =0. Q 1 2 u c 4 1 T =0 ( w → in (3.6). The standard non-extremal c c = 2 = − u x c 5 2 00 0 case discussed in appendix A, this corresponds 2 M in =0and for u − sinh 4 11 a 2 ∼ P a q 3 ,z ,z related by the zero-energy constraint (3.22) →∞ = y − 8 = 0. That would introduce an extra potential term in the z y 2 e 4 =0 b P 5 y + a, b, c 8 2 = 0 limit corresponds e b 4 P ,e = − 2 00 . That condition is satisfied only for a special choice of two 0 bu 0(sothat y b ≥ sinh 4 non-constant for f a, y = a, b, c y = 4 0 e x 0, we find → u In general, the system of second-order differential equations for From the effective “7-d black hole” point of view, these more general solutions To consider the non-extremal pure D3-brane case let us start with the general singular cases The discussion of this section applies both to We could keep 8 7 (3.20) has an extra free integration constant — an extra parameter of non-BPS with the integration constants and the following system: i.e. Assuming that 1as D3-brane solution we shall also set Φ = 0 and D3-brane solution but to a special non-BPS D3-brane solution. In this section wein trace the the previous singular section horizon to a problem similar of problem the in special the solution found 5. General non-extremal pure D3-solution and its regular and of the fractional charge doesin not the influence next section the that singularity this significantly. We will show certain non-extremal generalizations of the regular extremal D3-branez solution. deformation in addition to the constant correspond to thehas case non-vanishing when asymptotic charge. anscalar However, extra regular hair black scalar — holes otherwise (the shouldcharge we not radius is get have tuned a of to singular the zero horizon. that internal It we 3-torus) is get only asystem regular when of non-extremal this equations D3-brane extra (3.17)–(3.22) solution. scalar with free integration constants. horizon should be regular D3 with regular horizonOne [12] usually discards is such a more special general solutions case by of imposing a the condition more that general class of solutions. equations for the remainingto fields. having an Like extra the scalar Φ charge and most likely leads to a singular solution. JHEP04(2001)033 . 2 a (5.9) (5.8) (5.5) (5.6) (5.7) + (5.11) (5.15) (5.14) (5.12) (5.13) (5.16) (5.17) (5.10) 2 , q ) . p au , where the ··· 16 α = , , , − 2 ] a + 2 e 2 au ( γ ) 1 a. 4 + u 5 4 e 1 O 1 4 sinh − − = a + = 2 . dM = a ( x 1 qγ i 4 4 2 au ) 8 =2 ρ + = , − cu 2 q 8 e ¯ + q q 1 − au 2 z 8 e , ,z p − ,e ,ρ γ. dρ and ˜ + e − a au ) 1 = 8 2 qγ , k ≡ − α = − ··· (1 q ) au a g ) 1 e 2 [ ln + 2 c 2 − + u q cγ − / 4 1 u − ( γ 2 1 + ∗ ( cosh a 1 h z O a a 2 = = α c a, − + ∗ ,g . ˜ = = q 1+ 2 )+ + ) = h u c 4 i 4 ) ¯ bu qu a u 2 b sinh c 12 8 ρ ) sinh 4 q + a − − − a + q dX a 2 − e = i , c p 2 q ,z 4( = that the metric (2.4) takes the standard non- ) space. Then the constraint (5.3) is satisfied, ( b 4 a ,z 4( = − ˜ 1 ) q 4 1 =2 be ,ρ ρ e p dX ) a z − 1 2 + b, c q 4 4 a q au = + = 2 ρ 2 = = q 16 1 2 0 x = →∞ 4 − c 0) x − ,z p e − ck =1+ =1+ u ) ( ( z 4 +4 3 1 4 = ,e → x gdX y e O u 4 − 4 − ( =1 ( b 2 u q e e − / + au x z O 1 8 4 a, y = = = − − − a au + ,h e 8 h e 4 h 2 ak ln 2 ρ − 4 )wehave u e sinh 4 = = = e 2 4 1 ρ 1 ··· a y g h E = = 2 3 = + 2 10 non-extremal D3-brane solution [12] corresponds to the case when + y ∗ γ 4 a − 4 y ds ρ 2 e u (large au − u − is defined by ln 4 ∗ k 1 4 y =1 standard − = g y = i.e. to a line in the 2-parameter ( and (5.4), (5.5) become 5.1 Standard regular non-extremal D3-braneThe solution Then (see (2.5)) Note that near the horizon ( It is only for the choice while at large distances ( y At small where In an often-used parametrization extremal D3-brane form [12, 13] JHEP04(2001)033 ρ is ρ (5.21) (5.18) (5.20) (5.23) (5.19) (5.22) . , :itgrows au 8 2 6 u − e ds . . 2 / 4 = a, 1 q . Therefore, ρ ) . (One can check ··· 3 5 u to cover the entire ∞ = qu + 0. r u x , satisfying the con- a = 4 = a 4 → − ,g u − ρ z g 4 q . 2 au − / 4 2 e 1 ) 0, we still get the standard − 5 − e )+(1+4 ,b = i ) h → D3-brane solution (5.10) that au =1+ dM qu = 4 dX ( a satisfies the 1-st order equation i e 2 / 00 z 1 = G dX  ,h is the position of the horizon. x vanishes, then its location is at infinite au reduces to the standard extremal D3- 4 2 bu proportional to a ,h au 00 e c 8 =(1+4 standard b c − G x a + is not a monotonic function of e ··· 4 13 = 0 in (5.10) leads to the black hole in AdS 2 0; still qu , e 2 0 − of the solution [6] derived in section 4, has a )=2 ρ sinh 4 and − z k + 0 limit. The mass (density) of the solution is 4 and →  1 ∞ b 4 a − , having a minimum at finite dX =0where ( of the ρ e 2 h b 4 → + au = ∞ c ρ a limit 6 2 a = − 4 =1+4 − = = e z du ( 4 2 u c 2 =0 / − =1 / 5 ,and 1 , so this is the standard non-extremal black-brane type = P  − . ,ρ ∞ ) b ,h bu 4 ∞ a 0, but also u horizon. qu ) ρ 2 bu b b ,e 8 = ,g − → − =0and − a e (0) = u a>q bu g sinh 4 u 4( ρ 3 4 ··· − b  is divergent there.) be (large distances) and in the limit =1 + 1 regular 2 , the radial coordinate + =(1+4 = x 2 u sinh 4 8 u a 2 6 = 1 E − mnkl 4 e x (cf. (5.2)). Then from (5.4) we get = 2 10 + ds R b − − dX 2 i u K e 2 = ,K h − = − = 2 ∗ dX u p u + a aP f π ∗ g 2 2 + u 96 K = 2 P − 0 f + log changes sign, that means = dX u z C z (small distances), a novel behaviour. 2 4 4 au 2 8 − before we reach that singular point: − P u =1+ = − e e gives 1, i.e. 0 (0) = 0 to have the standard long distance limit. Explicitly, goes in the same log e z − 4 C z u z − .Notethat . Just as in the KT case, the derivative of f − u → e = = =1 ). the string coupling becomes weak and u a m z is similar to that of the 1 + 4 for large u 2 string u z − finite 4 ln 8 u e ds − 4 we reproduce the KT asymptotic behaviour P e approaches a constant for large u is such that − z 4 ∗ and then goes to zero. =0at C − f 0 e h u naked singularity Above we considered the case of This point is a curvature singularity. According to (2.4), (2.5), The equation for = Numerical analysis confirms that there exists a finite value of 9 0 = f ( which of course reduces to the KT expression in (3.14) in the BPS that then solution with a singular horizon at Since the derivative of where for small u so that the longitudinalthe volume horizon. stays finite but the transverse volume vanishes at while for large Now for large grows at small distances.opposite One finds case a of somewhat nicer behaviour of the metric in the so that is a string-frame metric becomes indicating the presence of a special point at finite behaviour of linearly with maximum, and then goes to zero at approximately becomes zero at JHEP04(2001)033 , , ) M + Nucl. (2000) B 382 , B 360 N -branes topology 3 (1999) 22 08 (1997) 65 3 SU( Nucl. Phys. S × , ) × On asymptotic J. High Energy 2 N , ]; S B 456 B 489 Phys. Lett. ]. Absorption of fixed × Nucl. Phys. SU( , ]. , R Phys. Lett. , Nucl. Phys. J. High Energy Phys. , , hep-th/9902125 Ricci-flat metrics, harmonic forms ]. hep-th/9803131 ]. (1988) 741. hep-th/0002159 Entropy and temperature of black ]. B 298 -branesonspaceswith . 3-branes on resolved conifold 17 ]. (1999) 270 [ (1998) 505 [ Black strings and p-branes ]; (2000) 123 [ The black branes of M-theory 2 Supergravity and a confining gauge theory: duality Brane resolution through transgression hep-th/0101043 Gravity duals of supersymmetric Black holes and membranes in higher dimensional Gravity duals of fractional branes in various dimen- On running couplings in gauge theories from type- hep-th/9602135 B 454 B 578 ]. Nucl. Phys. , hep-th/0011146 . hep-th/0012011 ]. hep-th/0010088 , hep-th/0011023 Phys. Lett. (2001) 086006 [ (1996) 3915 [ ]. ]; , Nucl. Phys. SB-resolution of naked singularities , χ (2001) 219 [ Anti-de Sitter space, thermal phase transition and confinement in gauge Finite temperature resolution of the Klebanov-Tseytlin singularity D63 D54 hep-th/9604052 (2000) 028 [ Adv. Theor. Math. Phys. c, G.W. Gibbons, H. Lu and C.N. Pope, c, H. Lu and C.N. Pope, , (2001) 103 [ hep-th/0101020 11 B 600 , hep-th/0007191 hep-th/9610172 hep-th/9903109 Phys. Rev. theories with dilaton fields gauge theories scalars and the[ D-brane approach to black holes cascades and (1991) 197. 052 [ IIB supergravity (1996) 73 [ Phys. freedom and confinement from[ type-IIB supergravity M. Cvetiˇ B 600 sions Phys. and brane resolutions Phys. Rev. theories [1] I.R. Klebanov and A.A. Tseytlin, [2] I.R. Klebanov and M.J. Strassler, [3] L.A. Pando Zayas and A.A. Tseytlin, [4] M. Cvetiˇ [6] A. Buchel, [5] C.P. Herzog and I.R. Klebanov, [7] S.S. Gubser, I.R. Klebanov and A.W. Peet, [8] E. Witten, [9] L.A. Pando Zayas and A.A. Tseytlin, 3 References [10] G.W. Gibbons and K. Maeda, [12] G.T. Horowitz and A. Strominger, [11] C.G. Callan, S.S. Gubser, I.R. Klebanov and A.A. Tseytlin, [13] M.J. Duff, H. Lu and C.N. Pope, [14] A. Kehagias and K. Sfetsos, JHEP04(2001)033 . ]. hep-th/9902155 , 18 hep-th/9812017 Two-boundaries AdS/CFT correspondence in dilatonic (1999) 39 [ B 449 Dilaton-driven confinement Phys. Lett. , gravity S. Nojiri and S.D. Odintsov, [15] S.S. Gubser,