JHEP11(2000)028 November 21, 2000 ) was recently generalized to Accepted: ). Here we present another gen- in a natural “asymmetric” way so hep-th/0002159 October 20, 2000, hep-th/0007191 hep-th/0002159 Received: ∗ D-branes, AdS/CFT Correspondance. The type-IIB supergravity solution describing a collection of regular and [email protected] [email protected] HYPER VERSION Also at Imperial College, London and Lebedev Institute, Moscow. ∗ Department of Physics, The174 Ohio West State 18th University Avenue,E-mail: Columbus, OH 43210-1106, USA Randall Laboratory of Physics,Ann The Arbor, University MI of 48109-1120, Michigan E-mail: USA ahler metric on resolved conifold and then solving the supergravity equations for that the 1-d action describingsolving radial the evolution resulting still 1-st admits order system. asolved” superpotential and The “deformed” and superpotentials cases then corresponding turn toThe out the to solution “re- have for essentially the theas same resolved in simple conifold structure. the case conifold hasIR. case, the The but same naked unlike singularity asymptotic the is UV deformed of behaviour conifold repulson case type is and still mayKeywords: singular have a in brane the resolution. the D3-brane ansatz with constanteralizing dilaton the and “conifold” ansatz (self-dual) of 3-form fluxes; (ii) by gen- eralization — when thecan conifold be is found replaced in byK¨ two the different resolved ways: conifold. (i) by This first solution explicitly constructing the Ricci-flat the case of the deformed conifold ( fractional D3-branes on the conifold ( Abstract: Arkady A. Tseytlin Leopoldo A. Pando Zayas 3-branes on resolved conifold JHEP11(2000)028 . 3 S =1 × 2 N S D3-branes )is 1 N , 1 T . The logarithmic running 0 r/r ln 2 ) M s g ( 2 c 1 + ) gauge theory and some key effects of intro- N N s g 1 c SU( × ) implies the presence of a naked singularity at small )= r ) r ( ( M Q + ,Q 4 N /r ) r ( Q fractional D3-branes on the conifold was constructed in [10]. This solution )=1+ r M ( The corresponding supergravity solution describing a collection of 5.1 Resolved conifold5.2 case Deformed conifold case 12 15 h (in IR from dual gauge theory point of view). A remarkable way to avoid the IR In [8] it wassupersymmetric argued SU( that the dual field theory should be a non-conformal ducing fractional branes wereand discussed, structure including of breaking the of logarithmic conformal RG invariance flow. and of the “effective charge” r singularity was found [11]:keeping the one same is D3-brane to structureansatz replace of appropriately. the the 10-d conifold metric by and the generalizing deformed the conifold 3-form To construct supergravity (andtheories string-theory) one duals is of interested lessp-form in supersymmetric fluxes “4 gauged [1]. + It 1dence + is compact natural [2] space” to by type try considering toe.g. backgrounds generalize D3-branes placing with the in them extra original at more AdS/CFTther conical general correspon- [7, singularities transverse 8] [3]–[6]. space byexploiting adding backgrounds, This the “fractional” idea fact D3-branes has that (D5-branes been topologically wrapped the developed over base fur- 2-cycles) space of [9], the conifold ( 1. Introduction Contents 1. Introduction2. Metric of resolved conifold3. D3-branes on resolved conifold4. Fractional D3-branes on resolved conifold5. Superpotential and first order system 8 3 6 12 1 has the standard D3-brane-typeby metric but with the “harmonic function” replaced JHEP11(2000)028 =1 parts 2 N S (resolved conifold) ahler metric of the (UV) asymptotic as × 3 , r 1 R = but has somewhat different , i.e. in the infrared. In [11] r 10 r M = 1 supersymmetry of such class of N 2 The type-IIB supergravity solution we find 1 (resolution). Here we complement the conifold [10] 2 (IR) behavior. The singularity of the analog of the S r (deformation) or by an 3 In section 2 we shall review the “smallIn resolution” section of 3 the we conifold shall [19] show and how find the geometry of One may discover this solution using two different strategies. One may start Alternatively, one may start with finding the Ricci flat K¨ The deformed conifold solution [11] has the same large It is of obvious interest to explore further the class of backgrounds which have Various aspects of branes on resolved conifold were discussed, e.g. in [12, 13, 14]. S 1 resolved conifold (which, as far asin we an know, was explicit not form) previouslybrane and given in ansatz then the solve with literature the constantfractional type-IIB dilaton D3-branes. supergravity and equations 3-form for Ascases, fluxes the in representing the D3- the the complex inclusion other 3-formproperty of was two field emphasized (“standard” turns in and out [11, “deformed” to 16] and conifold) be the self-dual. The importance of this backgrounds was recently proved in [17, 18]. the corresponding Ricci flat metric explicitly. coincides with the original background(though of still [10] for singular) large small solutions of [10, 11] in the resolved conifold casewith was anticipated the in “conifold” [16, ansatz 17]. very simple for and the natural way 10-d by background allowing an in “asymmetry” [10] between the and two generalize it in a in the metric andthe in spherical the symmetry NS-NSequations as 3-form from in introducing a [10] 1-d two actionin new one describing [10], functions. can evolution the in then potential Assuming radial termis obtain direction. in true the this Remarkably, also as resulting action in cansupersymmetry supergravity the be of “deformed” derived the from case resulting a1-st solution of superpotential. which order [11] This follows and equations. then isresolved by conifold. In consistent solving with the the resulting expected process, one explicitly determines the metric on the and deformed conifold [11] solutionssponding by constructing to explicitly the the resolved background corre- conifold case. changes in the presencesolution of D3-branes, [20, i.e. 21] find in the analog case of when the the standard D3-brane transverse 6-space is replaced by the resolved the original conifold onesome [10] desirable but properties of is thiscounterparts regular background of at were the small established, existence includingdual the gauge of gravity theory. confinement and chiral symmetry breakingsimilar in “3-branes the on conifold” typeof structure [15]). (potentially Given including the also topologying the of out solution the the base space, singularity therean at are the two apex natural ways of of the smooth- conifold. One can substitute the apex by JHEP11(2000)028 can 1 (2.1) CP .Thechoice 2 S . bundle over  2 C VY XU  ≡ ,  4 2 =0 iw . It seems that the averaging over iw 5 + S UV − 3 1 × − w w 5 − 3 This equation can be written as XY 2 4 3 AdS . factor and is singular. We shall compare this iw iw 5 =0 + + 2 i 3 1 i.e. AdS w w w 2  =1 4 i 2 , 1 P √ : 4 = =0 C W W det = 1 supersymmetric RG flow in the dual gauge theory. In section 5 we shall explain how the same solution can be obtained from a In section 4 we shall generalize the D3-brane solution of section 3 to the presence N We are grateful to I.More Klebanov details for on this the suggestion topological and structure of explaining the the resolved relation conifold to as [13]. a causes a singularity (present also in analogous D3-brane solution with deformed 2 3 2 of point corresponds totheory, giving breaking expectation gauge values symmetrywas for non-singular and scalar in conformal fields IR, invariance. in approaching the The dual solution field in [13] D3-brane solution with the one ina the radially case symmetric of the solution deformedthe corresponding conifold apex. to [11]. 3-branes We In consider smeared [13] over a the 2-sphere 3-branes at were instead localized at a point on conifold as transverse space). 2. Metric of resolved conifold The purpose of thisresolved section conifold is following to theto detailed write introduce discussion down in explicitly a [19]. globally themetric well-defined Ricci Though on metric flat it each on is metric of not thefollowing on the possible quadric resolved the two in conifold, covering one patches. can find The a conifold can be described by the conifold (the coefficient in thetrast metric to is the a D3-brane harmonic on functionbackground the on does conifold the [5] 6-space). not the have In short con- an distance limit of this supergravity S 1-d action for radialof evolution 1-st admitting order a equationsexists superpotential, as also i.e. in in [10]. by the solvingprocess, We a deformed shall making system conifold point simple out case1-st ansatze that of for order a [11]. the systems similar 6-dthe associated superpotential As explicit part with forms we of of the shall the theway. Ricci-flatness demonstrate resolved metric Identifying and in explicitly equations and deformed the conifold the allows identifying 1-stgrounds metrics the one order in of system a residual to straightforward (whose supersymmetry) find existence isa is useful potential expected correspondence for between generalizations on the and the the radial for evolution establishing on the supergravity side and of fractional D3-branes onsolution the and resolved conifold. show that Wesingularity, it so shall one has analyze may the a hope limits short-distance that of singularity. it the may This be is resolved a by the repulson-type mechanism of [22]. be found in [19]. JHEP11(2000)028 ). )is 2 (2.8) (2.4) (2.6) (2.5) (2.9) (2.7) (2.2) (2.3) ,λ 1 . λ 2 X, Y, U, V dF dr ahler potential ≡ 0 =0bythepair , = 0 is the conifold 2 a , where for the metric dγ UV dr 6= 0. Working on this ¯ . n ) . ,F m 1 − ≡ 2 g ) | λ 2 0 0 ) ahler potential. In contrast Y 2 F | | , 2 2 | XY λ ) r ahler potential is of the form | + 2 ln det | dλ , 2 + | | λ ¯ . 6= 0 and thus the general solution can n . ,γ | 2 0 ∂ 2 U a | (1 + λ  m F =0 2 2 =0 ∂ )( a 2 r )(4  4 | − 00 r 1 2 λ ln(1 + ≡ | (any pair obtained from a given one by F λ λ +4 = 2 2 − . 1 2 r a ¯ n 2 4 | Uλ U Yλ Y 2 ) m γ 4 + − −  /λ CP 2 0 R 1 in the region where W K,whereKistheK¨ a  λ )+4 ¯ d n F ∈ ,γ 1 2 † ( ∂ 2 )=(1+ = 0 r ) = +6 r ( 2 m /λ µ F W is the “resolution” parameter ( 3 2 VY 2 3 XU W ∂ F † W ,λ γ λ a = tr( 1  = | W λ ¯ n = 00 ¯ )= n where m 2 K= and m F λ , g a g 2  r = tr( ahler metric is is allowed to be zero we have )+ det +4 2 Vµ Xµ 6=0. 1 r − γ − 1 λ W ( λ d γ † 0 V X γ  W d = ) are the three complex coodinates characterizing the resolved conifold tr( 6=0.Notethat( W 0 2 F is a function of λ 1 = λ U, Y, λ F 2 The conifold metric is In the region where ds 4 which is integrated to give uniquely characterized by Following the analysis of [19],overlap based region, on one the concludes transformation that of the the most coordinates general in K¨ the The Ricci tensor for a K¨ to the the casesis of not the a conifold globally or defineddefined quantity, the by and deformed is conifold, not a here function the of K¨ only the radial coordinate where case). Thus the metric is in (2.6) The Ricci-flatness condition implies multiplication by a nonzero complex number is also a solution). Thus ( where patch a solution to (2.2) takes the form of equations The resolution of theResolving conifold the can conifold be means naturally replacing described the in equation terms of ( in the patch where Thus ( be written as JHEP11(2000)028 = 3 / 1 (2.10) (2.11) (2.14) (2.15) (2.12) (2.13) (2.16) 1) . . + − in terms as a new ) ) , 2 , 3 4 γ / 2 r ! 2 W 2 4 i θ 6 , − a . r dφ
( i 32 cos 2 =1 θ O 2 φ 1 e 2 − θ + 8 cos 2 + r a 2 sin 2 2 2 θ √ =1 ) ,i i X e 2 i . − + φ a 3 6 + ) + / dφ (0) = 0 (as should be true a 2 1 4 i φ a γ r θ one uses the cubic root ( dψ 16 − 6 3 ψ a (

− +4 ) part of the metric shrinks to 2 i 2 )= 4 2 1 γ r r √ 0 =sin ( ( re 4 ,φ γ i 1 4 2 1 1 φ < 1 4 = →∞ 2 + ≡ ) correspond to the two SU(2)’s. r + r 2 ψ,θ
) γ. ( [19]. Note also that ( 1
r ,ψ 2 3 2 φ 2 i ( 3 2 3 e / 5 it is very convenient to consider ,e 4 S i ≡ dφ ,φ ) r ,Y i + 2 2 2 θ θ a dθ 2 1 ρ ,γ 2 = 2 θ 2 2 θ the 7 r +4 e ) The real solution is = 2 γ γ 6 ,N ( r . i r ) 5 γ θ 0 provided for γ cos ( ) 3 r )and( 2 2 +sin 4 1 ( stays real). 1 1 O > 2 3 θ 2 i / = γ 2 dφ + + 1 ,ψ r 0 ) part stays finite with radius 2 1 4 dθ 2 2 ψ ahler, Ricci-flat metrics. γ θ N r introduces asymmetry between the two spheres. Defining cos ,e e ,φ 2 4 ) i 2 1 ,φ a . 2 a 2 θ r 2 =1 )+ 1 0 φ i 2 θ X and dφ r 2 γ + θ 72 ( i = 0 we have ( 1 γ θ 4 1 3 +sin φ 2 / a − 4 1 dγ SU(2) symmetry 2 1 + 2 2 S ,and( + tan ψ − r 2 cos + × 2 2 dθ 2( ψ a ( 2 N )= iφ i/ 6 2 2 4 2 1 =1 dr + i 0 − a X becomes complex, a r dr √ 1 0 re e ( γ ψ + γ d + N 0 +4 = = = γ = 2 = 2 6 λ a = 0 case of the conifold). dψ but are nevertheless K¨ U 0) = ψ 2 6 2 has SU(2) ds 1 a , = (while − 1 ds → T ψ Since 3 W Here One could keep the integration constant in (2.9) which would lead to more generalThis geometries. expression applies for all r = e √ 7 5 6 ( i 2 γ γ 1+ radial coordinate introducing zero size while the In the conifold case To write down the resolved conifoldof metric the explicitly two we sets will of parametrize for Euler angles, exploiting the fact that the resolved conifold solution the metric can be written as We set the integration constantin to the zero, assuming that As follows from (2.11), for small These more general geometriesover are compatible with metrics on the conifold that are not the cone Then the resolved conifold metric takes the form Note that the parameter the veilbeins JHEP11(2000)028 U(1) (3.1) (3.2) (3.3) (3.4) (2.19) (2.17) (2.18) (2.20) / ). Using , ρ
( 2 h SU(2) 2 φ , e = × . . . +
a h  2 2 one can construct
2 θ 2 φ 2 e e 2 φ goes to zero or when e mn ) + g 2 a 2 Φ=const + a =SU(2) 2 θ , 2 e part shrinks to zero size 1 , 2 θ ,  e 3 1 n +6 2 2 2 S T a a 2 , + ρ dy 9 ρ 3 1 ( + . m 2 φ 1 6
e dx 1 dy 1+ ) 2 φ . + + ∧ e y  2 2 1
2 ( )=0 2 θ 1 a a + h =0)the ln e 2 φ dx n mn 1 e 4 4 2 θ ρ g ∂ a +9 +6 6 1 ∧ ) e L + y 2 2 1 2 mn 81 1 6 ( + ρ ρ + 2 θ 2 / e dx gg 2 ψ 2 ψ − 1 ≡ e we get e h ∧ √ 2 2 9 1 ) ( 2 4 0 ρ ρ 2 ρ θ  + 2 ρ L ( 6 1 m 2 a 2 µ dx ∂ κ 6 1 ρ 9 sin + g ∧ dx 1 1 + + 2 ψ + √ 1 µ When the resolution parameter θ 2 e 2 ) part approaches finite value equal to − 0 2 8 2 dx h ρ dρ SU(2) invariant form of the resolved conifold metric which dρ ) ) dh 3 2 ,φ y )sin ) ρ = × 2 ( 2 = ( ∗ θ 2 a h = κ / ( 1 9 1 0 2 − →∞ +6 → S ρ h + ρ ) 2 ) 2 6 2 ρ 2 6 = =(1+ ( 3 ds dρ 5 ds ( ) 2 10 ρ ( F ρ 1 ( the metric (2.17) reduces to ds 108 1 ρ − = is a harmonic function on the transverse 6-d space: κ it reduces to the standard conifold metric with g h = √ 2 6 It is easy to check directly that this metric is indeed Ricci flat. 8 ds →∞ For small where Let us solve (3.3) for the resolved conifold metric (2.17) assuming while the radius of the following generalization of25]) the standard [20, 21] brane solution (see, e.g. [4, 24, 3. D3-branes on resolved conifold As is well known, given a Ricci flat 6-d space with the metric This allows one to avoidregions. the issue of Then howsimply using to as define (2.8) the and expression (2.9) (2.10) in the different resolved conifold metric can be written ρ as the base [19, 23] where we shall use in what follows. This is the explicit SU(2) This shows once again that near the apex ( that JHEP11(2000)028 and , (3.5) (3.6) 2 (3.7) (3.9) (3.8) , (3.10)  rather 1 /ρ
θ 2 2 , 2 e a 2 φ 2 S i e ) + √ 2 4 =9 1 + g  . x 2 2 θ 2 + 0 solutions. e = 2 3 g 4 2 ( space (see [5]). ψdθ 0 (or, equivalently, x> y 2 1 2 τ , a 1 → sin 2 T a + − ,g
× 2 1 . Introducing 1 φ r 5 2 φ , e dφ e 2 , 4 2 2 θ − )+cosh √ a 3 L + which has no / 2 AdS 2 9 . 2 2 1 1 =0.Itiseasytocheckthatthe g 2  2 θ  4 c 4 sin e ρ τ = ρ + L = = ψ 2 2 + 1 − 3 x τ + g 2 ψ ) ( 0 cos − e τ h ) 2 τ x ≡ 2 sinh 2 ,b y 7 2 ,g 6 1 )= 2 2 1 b ρ sinh(2 9 θ + 1 2 e 0 2 2  → − →∞ , √ ρ )+sinh dy ) 0) = 1 2 2 5 now blows up at ρ 2 6  3 8 we get g ( )= a h τ → − ds ρ + + ( ψdθ ( ρ 2 µ b ρ = K ( h 2 dτ does not vanish at any real value of dx + ( µ ) = 0 becomes ln(1 + 1 h µ +cos r − ( 2 dx ) h dx 3 2 2 µ b dφ y ,g defined in [11] are K 2  1 dx θ n (3 φ b g b ρ e h 2 and we used the expression (2.20) for the short-distance limit of the sin K + √ = = ρ 3 ψ 2 / , part of the 10-d metric still shrinks to zero size but the size of 0  4 √ ψ  3 → the equation ρ − e sin 2 1 S = 0 ) is the deformation parameter, = 0 is the curvature singularity (while the Ricci scalar vanishes as for any h 4 4 = = ≡ 2 10 ) limit the solution approaches the standard flat space or conifold one y  L = ρ a 2 ρ 1 5 1 2 6 81 ds  For completeness, let us compare the above solution with the one in the case The g g ( It is easy to check that 9 ds = 2 where For small values of the radius so that the 10-d metric becomes than approaching the constant point D3-brane solution of the type (3.2),be the compared Ricci with tensor one is singular). inthe the This short-distance case behaviour limit is of of to the the D3-branes geometry at was the regular conifold singularity where and the 1-forms when the transverse 6-space is the deformed conifold with the metric [11, 19, 24, 26] where resolved conifold metric. where we have chosen the integrationlarge constant so that in the c JHEP11(2000)028 will =0 part (4.2) (4.1) 5 ρ (3.12) (3.11) 2 F S . and 0 3 . ρ F → →∞ . ] 2 φ ,τ e , 3 0 , ∧ ,τ τ/ , 2 3 → 1 4 2 2 6 ρ θ Ω − − ) e . The ansatz for the R-R e d τ 2 6 ) 2 ds 1 1 3 ) ρ f / h h ( ds ρ 3 1 0 , ( − 3 2 ( / 3 / 2 4 / f . In the 10-d metric the 2 ρ 3 2 / 2   φ = 0, i.e. the forms 1 m part blows up at the point 4 = e √ + (12)  3 h 3 1 1 4 3 ∧ φ + f + S + dF 2 e 1+  ν 2 2 . In the short distance limit of the 6-d θ µ 2 e ∧ Ω /        ) dx 5 dx d 1 ρ µ θ 2 − µ ( 8 = e ρ 2 ) 2) dx f 3 2 1 dx ρ / we get / ) ( 2 µν 0 + ρ 1  + τ η (3 ( f 1 τ 2 2 2 [ ρ φ / − m e 1  − ∧ √ dρ we recover the D3-brane on the conifold limit with 4 parts in the resolved conifold metric (2.17) − ) ∧ h L τ 2 τ = = dρ 1 dτ θ S 0 = = e = 2 10 ) → 2 2 ρ 2 10 ρ in (3.1) is then found to be ds ( ) m 1 sinh(2 2 6 ds h f dB 1 2 ds  for large ( = = 3 2 3 ,and Z τ/ τ 1 B H e 6 h / . For small values of 1 3 ∼ 4 3 / − 2 6  / ρ 0 5 /ρ will be the metric of the resolved conifold (2.17). Our ansatz for the 2 4 h is dictated by the closure condition L 2 6 3 ≡ F + ds )= ρ 0 τ ( h h = The “conifold” ansatz [8, 10] corresponds to still shrinks to zero size but the radius of the h where 3-form 4. Fractional D3-branes on resolved conifold Let us now study athe generalization of case the of D3-brane additional solutionof 3-form of [10] fluxes, the describing previous with section a the to the collection aim of case to regular when find and the themetric fractional conifold analog will D3-branes is of be on replaced the the the by solution same conifold the as in resolved in conifold. (3.1), The ansatz for the The harmonic function where NS-NS 2-form will beasymmetry a natural between generalization the of two the ansatz in [10] motivated by an deformed conifold metric the 2-spherefinite shrinks to radius zero related size to while the the 3-sphere deformation part parameter has which is the curvaturegeometry singularity. is singular. As inD3-branes That [11] the is one why resolved needs to conifold tofor get set case, the the a 5-form the “bare” regular field D3-brane near-core (and solution charge thus to after for zero adding to the fractional make Ricci possible tensor) to vanish at small Introducing JHEP11(2000)028 ) 3 3 s K. F H g 1 11 (4.3) (4.7) (4.6) (4.5) (4.4) (4.8) (4.9) 108 = . Φ →
, e 1
φ K 2 e φ , e ∧ P 1 ∧ 2 θ . 2 √ e 1 2 0 θ 2 φ . 18 e f =0. e 2 ) 2 3 2 2 Γ − → ) ∧ F a 2 (modulo an irrelevant · =0case. 2 − P a θ 3 1 a e 1 +Γ f , +9 H φ 2 ∧ e ) 2 +6 . − , φ 2 ρ 1 e 2
) ∧ ( a , i.e. using (2.17) we get φ 2 ρ = 2 3 2 e 3 1 ∧ ( − a ρ θ 2 F 2 e f ∧ PK , +9 θ P 6=0). s e 3 1 2 2Φ s PK +9 0 g θ 1 e a g s ρ ∧ e 2 f ( , 1+Γ g 3 dx . 3 3 ρ ) = ∧ ( 324 2 − ρ 1 ∧ ψ 3 2 2 2 a 3 φ dx ψ e 2 324 2 ρ h e = e P ρ . 2 2 H ∧ ) 2 s ∧ 2 0 2 ρ h dx − ∧ +6 ρ g 2 k 3 9 is satisfied automatically, and from the ( 9 1 2 ∧ θ = = =0 3 ρ dx dx K e 1 10 0 0 0 = 2 H ∧   ∧ f ≡ − = 2 dx 0 ) ) 2 ,f 2 1 . 2 ∧ 2 2 2 Γ f − 2 ∧ φ T 5 F a a 2 dx 2 one obtains the following three equations ( e a dx 0 F − 2 1 √ 3 ∧ 6 ∧ ∧ Γ +Γ ρ +9 +9 dx F 0 0 +9 hρ 2 1 1 2 2 θ = +Γ )= hρ , f ∧ 2 ∧ ρ ρ e 3 here are related to the ones in [10] by 2 2 dx dx ρ ( ( 0 ( 5 1 f F 0 0 2 1 f dρ F ∧ K P ∧ ∗F − f f ∧ ∗ 2 s − 0   ψ g = h Φ + dρ ) e 1 2 dx ( h f Pe F ) 2 a d )= =3 2 a 3 K 0 1 a = = f H +9 ρh 3 5 Pρ ∗ +9 108 2 3 F F 3 +9 ρ 2 Φ and function ( 2 ρ − 3 ρ P e ρ ( − ( d = = = 3 3 F H ∗F The constant dilaton condition implies As in [10, 11] we shall assume that the dilaton Φ is constant. Then the ∗ As in [10, 11], the axion equation is satisfied automatically since Note that our definition of the basis of 1-forms differ from [10] by numerical factors, so that ∗ 11 10 Then using the metric (2.17) the 10-d duals of these forms are found to be It follows from (4.7) that for Γ = 1 one should have constant) which was precisely the assumption of [10] in the is the ratio of thedifference squares from of 1 the is radii a of signature the of two the spheres resolution in ( the metric (2.17) and its Here Combined with (4.7) that gives equation of motion equation be taken in the same form as in [8, 10] the constant Also, in the case of [10] JHEP11(2000)028 are P (4.12) (4.10) (4.13) (4.11) (4.15) (4.14) (4.16) )), i.e. and 2 a Q , 2 3 , +6 F . 2 s  . ) ] ρ g gives 2 ( 5 6 1 ), i.e.  ) 3 3 f 2 2  3 , ρ F − 5 a . − F )  ∼ ∧ 2 Φ ) 1 2 2 e )] a f 3 g a +9 2 )= ( . 2 a ) which is not very illumi- H 3 2 + √ P ) ρ ρ +9 , F 2 3 ρ +9 ( ( s = 2 2 ( 2 8 + h +9 a a g 2 H ). The constants ρ (1 + Γ 5 ρ ρ 2 ( K a Φ ρ ) it is sufficient to consider the Q +9 +6 + 8 1 ρ 3 − 2 2 2 ρ dF ( ρ − 2 ρ ρ ( e 3 =  ) P ( ρ/ h 16 = 4 supersymmetry of the resulting 2 s +ln[ = H , 1 ρ 24 g a 1 − ln K 5 2 ) ( )= − s ,d 2 2 a 2 F ln[ ρ − g = a ρ ( 324 ( +9 ∗ are dual to each other in the 6-d sense. 18 2 Li h − κ 2 − =1 2 a 1 d 3 10 12 ρ 2 +9 ∆ ρ − of regular and fractional D3-branes. ( = F 2 a 1 2 = 18 2 3 i.e. − N ρ 0 ρ Q − ρρ −  36  M 3 0 ρ g 2 h ln(   and )= = ) P h ) 2 , s 108 3 P P ρ 2 R and g a s s ) ∂ 0 a − H 2 g g 2 3 1 f ρρ 2 6 3 1 N P = +9 s − +9 − 0 gg 2 g 0 2 h 1 ρ Q √ ρ ( )= )= f 36 ( ( 3 ( ρ ρ 3 ρ ρ ( ( P ρ 1 2  )= ∂ f f ρ g − = ( 1 0 √ = K 2 K 0 / 3 h − h is the 6-d metric (2.17) ( mn g The Bianchi identity for the 5-form backgrounds [17, 18]. This property, together withresolved) conifold the metrics Calabi-Yau implies nature the of the (original, deformed or Integrating this equation we get The symmetries ofcases the [10, metric 11]) ansatztrace that imply of to the (again, determine Einstein as equations, the in function the other two conifold where we have chosenFrom the (4.9) we integration find constant (we to omit be trivial constants related of to integration) the one in (4.10). It is easy to see fromconifold the above cases relations that, [11], as the in the forms conifold [10] and the deformed where Note that (4.13) and (4.15) imply that and thus from (4.10) proportional to the numbers Integrating (4.13) one cannating find as the it contains explicit the form special of function JHEP11(2000)028 con (4.19) (4.18) (4.17) (4.20) . ) . Expanding 2 h ρ, , ρ ρ (˜ 2 ln ) (4.15) vanishes a O 2 h ρ 2 2 ( ρ P P + ρ s s K g = 4 g a 6 ρ . 2 2 +6  P − Q s 1 Q g Q 2 − = = 2 + ) the singularity is located at − at short distances should imply 0 Q 0, as naively expected in the limit of Q h 4 h a → 2  , 0 )= ,K P ,K h ρ s 1 P 4) 2 g − / 0 +˜ h Pa 3 K 11 Pρ s +1 ρ s ρ ) the solution becomes g g ( . ρ a 3 2 3 12 1+2 − P before reaching the singularity at (ln − 4 s p 4 2  ρ g = ρ  K P = ρ 0 as we use the Einstein-frame metric). ,h 2 ρ s 2 18 0 2 ) g a 1 Q 2 0 = − s 3 − . ρ h g (˜ h ρ 2 ) limit we reproduce the solution of [10] with its charac- O ρ a Q = = ,f 2 + 162 3 6 >ρ 1 0 2 h a + 4 ρ, f ρ − h h ˜ ρ L  2 + ρ Atthesametime,thefive-formcoefficient 13 P a K 2 Q ρ s 0 Pρ + ( 3 ρ s 2 g h 12 √ 0 g . (and ρ h 2 = = we get = a implies the presence of a naked singularity at Q )= 0 1 1 = =3 h K K ρ − f h 0 1 ρ ρ h Q f +˜ 2 , =27 = K P K 4 ρ s ρ ρ ( L g = K In the short distance limit ( One may expect that this naked singularity may be resolved by the enhan¸ If a mechanism similar to the one in [22] does apply in the present case, then the In the large Reference [22] used the form of the effective action for the D6-branes wrapped over on K3 One obtains the same value by simply sending =3 ρ 13 12 2 h For small number of fractional D3-branes ( ρ around The form of where that probe the geometry.been extensively The discussed in caseYau the of of literature D6 [28]. dimension on In 6than K3 the the and is present whole the case similar space. weChern-Simons D5-brane to are Thus term. we the dealing we The are case have with Dirac-Born-Infeld non-trivial a of dealing part tangent Calabi- of D4 with and the on normal here action K3 bundles is is also which which wrapping different will has (see a affect [29] the two-cycle for rather details). mechanism [22] (as wascase). originally expected First, [10] thefeature for singularity is the is the singularity offootnote underlying in the 2]). SU(2) the right conifold symmetry repulson Tohowever, of make type non the the trivial. [27]. argument 6-d for Second part such important of resolution the at metric a (see quantitative level [22, is, This is similarparticular, to the the constant IR value of behavior the found warp factor in the deformed conifold case [11]. In geometry should “stop” at teristic logarithmic behavior at small radius. again confinement in the IR. JHEP11(2000)028 (5.2) (5.4) (5.3) (5.1) (5.5) ). To be able , )+ ρ 1 2 which measures 2 φ φ e y e + ∧ , 1 2 2 θ θ  e e ) ( 2 y ∧ 2 φ + e , 1 x is related to ] φ e 2 e + φ u 2 . Note that the metric of the e + ∧ 2 θ u e 1 2 ψ ∧ ( θ e y 2 e x , θ − − e ) x ∧ p was set to zero in the “symmetric” ) 1 e 6 . φ ψ u − y u e ( e + 0 e ) 2  ∧ f u . ( 1 θ + 12 )] K e )+ 1 u 2 parts ( φ ( will be considered as independent functions − = e 2 2 du f 2 2 ∧ φ f S F e + 1 − θ ) µ e ∧ u ) 2 ( and dx u θ , 1 ( µ e 0 f 1 1 ( [ f f dx [ ∗F P ∧ A ∧ 2 ψ + + e ( x du Pe F Q − p = = = ≡ 2 e ) 3 3 5 are functions of a radial coordinate u F F H ( = 14 K 2 10 ds ). u A, p, x, y coming out of the p-form part (5.2)–(5.5) of the ansatz. We shall assume As we shall explain, the same strategy applies also to the case of the deformed In this section prime will denote derivatives over u 14 where previous section (4.1) and (2.17) belongs to this class ( conifold ansatz [10]). The ansatzand for (4.3), the i.e. remaining fields will be the same as in (4.2) conifold ansatz considered inthe [11]. same The structure corresponding as superpotential inthe has the 1-st conifold essentially order [10] system and found resolved in conifold [11] case, and thus reproduces checking5.1 its Resolved consistency. conifold case Let us choose the 10-d metric in the following “5+5” form Let us now demonstrateof the how previous the section 1-stresolved can conifold order be metric. system derived We directly shall ofan without follow ansatz equations the using for original and the the approach expression thecompute of 10-d for [10], the solution metric the i.e. 1-d and start action p-form with forgravity fields equations the which of radial has motion evolution the restricted thatsuperpotential required to reproduces and symmetries, this the thus ansatz, type-IIB obtain show super- a that this 1-st action order admits system. a 5. Superpotential and first order system Here we have explicitlytity used for the constraint the (4.10) 5-formare following field satisfied). from (so the Bianchi that Thus iden- the only Bianchi identities for all three p-form fields that the axionΦ=Φ( is zero (this is consistent with (5.3)) but will keep the dilaton of an “asymmetry” between the two to describe the resolved conifold case we have included the function JHEP11(2000)028 − (5.9) (5.8) (5.6) (5.7) 2 , ) (5.13) (5.12) (5.11) (5.10) ) + C 2 , in (5.5). y ∂ (  2 ) ∂B 2 2 2Φ 0 2 ,f e , ). As follows 1 C f 2 1 2 y f cosh 2
− 2 -term in (5.6)) 2 − x ,f e  2 1 10 2 − 2 K 5 − + x R , F 2 Φ ,f 4 2 ) ∂C e 0  − 1 Φ 2 2 5! 2 ) e f 1 · , B y P ϕ 2 ( 4 + ∂B 2 .  − ( V . ) e Φ − p ( +5( 8 ϕ − ··· x, y, p, x 2 − ( e e 4 2 ) . b 1 8 2 0 + − 1 2 W =( 12 Φ ϕ 2 )=0 ∂C 3 1 a + − a [33]. The 1-d action reproducing 0 ∂B ϕ 0 W 2 e − ( = y ϕ − 5 is a combination of ϕ f ∂B 3 2 1 ) 2 2 5 V F − P −C = ϕ ∗ − K ( Φ) 0 + 2 ∂C = + b cosh ∂ ab 4 b = in (5.7) can be represented in the form 0 0 ( 2 1 x 0 5 C ∂C 2 G ϕ 2 1 ∂ϕ ∂W ( V Φ a − F 2 1 10 ,F 0  a p Φ  13 ] − 2 ϕ e 2 4 1 ... − e ) ,A ∂W ∂ϕ 10 1 2 ϕ C b 12 ,f 0 . − + ( [ ab R (3!) A ∂ 2 y ab − 0 · ∂ϕ ∂W G 3 1  5 p =0 G  1 8 ab 4! 10 2 1 ≡ y A · g G 4 = +6 ... 2 2 1 − − cosh 2 2 ) 0 V 2 x 4 0 √ y − 4 = du e 2 1 − A  a p ∂C 0 3 4 ( x Z ϕ + − c 10 2 e 0 d 4 1 x = , Z S = ] . It should be supplemented with the “zero-energy” constraint .. 9 b 2 10 )= 2 10 0 1 B κ κ ϕ . ϕ 2 Vol [ ( 2 a 0 ∂ 4 V − ϕ − ) =3 = ϕ = ( ... c 10 ab ) S 2 G The type-IIB supergravity equations of motion follow from the action Similar expressions corresponding to the case of ∂B ( In this case theare 2-nd satisfied order on equations the solutions following of from the (5.7) 1-st and order the system constraint (5.8) from the “matter” ones and it is assumed that The existence of asee, superpotential e.g. (usually [30, associated 31] but with also residual [32]) supersymmetry, means that supplemented with the on-shell constraint where from (5.6) in the case of the ansatz (5.1)–(5.4) where we separated the gravity contributions (coming from the the resulting equations ofgeneral motion structure restricted to the above ansatz has the following In our present case we have 6 dynamical variables JHEP11(2000)028 W is set (5.20) (5.19) (5.14) (5.15) (5.16) (5.17) (5.18) K , , so that the 2  . f c y, )] ± 2 − f 1 . Also, the poten- y sinh . f 2 0 2 − x f 2 1 ≡ , − f =0 p ( A = ± 2 y 0 cosh 3 f − P 3 2 +2 Φ 1 3 0 e 1 = 0. The 1-d action of [10] x f + − y − p = sinh 2 0 Q y (resolved conifold) should be a K, K [ e , x x x K, 2 2 2 y × ≡ x − 2 − − 4 2 p p p 2 =0if =0if − cosh 4 − 4 4 / R p p e y, 1 e e  0 + y 4 3 1 6 − 2 1 1 2 2 f e = b =0imply − Φ+4 K, y 1 6 − x + + = 10 2 K Pe y + y y 14 q − sinh p M − − y 4 2 ,h e b x = with = 0. Indeed, as follows from (5.15)–(5.17) the 1 2 cosh cosh cosh 2 0 2 = x − x x cosh part in (5.1) satisfies 2 p 2 2 − from the action using its equation of motion and x K 4 x 4 − 2 − − ,e 2 y p p p + q . Since the superpotential (5.11) is the direct sum R 2 is satisfied by − 2 2 i f e p x, y, p − − − f 4 2 y e Khe e e ,f − − cosh 3 1 ) y e − + + x y 6 1 2 +2 p p − y, e = p 4 4 − p 0 p − e e x. 4 2 h p e − Φ+4 4 (sinh = − 3 1 e coth 3 e )= p b 3 1 − − Pe 6 − ϕ ( 2 is satisfied automatically by that multiplies ======2 ≡ =0. 0 0 ,f 0 0 + W 1 A 1 q p x f y f A dq dy dy +2 dx ,f x (which at the end should be the same as in (4.1)) is constant if Φ − p h 2 e To establish the equivalence of this system with the one found in the previous It is quite remarkable that, just like in the “symmetric” case considered in [10], then setting solution to these equationsfactor with of the gravitational and matter terms, so that equal to 0. The equations for equation for tial (5.12) depends only on one of the two combinations Note that the dilatonspire factors so in that the theΦ kinetic superpotential = (5.11) does const and not on dependA, potential the x, y, on (5.12) solution p, the terms of dilaton. the con- resulting This 1-st implies order that system of equations (5.10) for appeared in [10]. Indeed, thatand restriction was (5.12) consistent. the As follows equation from (5.7), for (5.11) may be found by eliminating We see that (5.13) correspondingsolution to of the this “standard” more conifold general case system. is indeed asection, special it is usefuldepend first on to look matter at functions the “gravitational sector” equations that do not the more general systemgiven by (5.11) the and direct (5.12) superposition still of the admits gravitational a and simple matter parts superpotential JHEP11(2000)028 , y 2 du e x and 2 y − which (5.21) (5.22) (5.23) = p ). The 2 y . u . − −  e x ) ) 2 2 4 a /e g 3 we finally = to get simple y / + + ,f,k,F u dt , x +9 2 3 e Φ 5 2 g , =4 g ( ρ t + y ( ∧ 4 5 − and ρ g x 4 . One then finds that ,c y e g 2 x, y, p, y ∧ − 1 a z ∧ are constants. As in the 4 324 2 1 2 instead of g 3 )+ − 2 g 2 =( Q − ρ e = , ∧ cosh g , ∧ a x 3 3 2 ] )] = = , + g 2 4 + ϕ , i.e. on the equation for u p g y and g ) 2 1 )] ( + 6 4 b K + g u − z ∧ g ∧ F x ( ( P 2 y 1 3 and e ∧ F /e − + g g x ) x ) 2 − A e − g − P u u p ( ( = 0 6 ,e + P + )[2 k − K ) 2 5 3 2 e u g g 15 ( + a = and introducing x f +[2 2 ∧ 5 − g 5 p 1 +6 6 F g g ∧ )+ − 2 ( ydu y, dz/dt ∧ e 1 ρ u  ( ( g ∧ 2 2 ) g F ρ , u ) sinh ( du ∧ ∗ )+ 5 0 u 1 ) 2 x 36 1 ( f 2 F u =sinh [ g is thus one of the three integration constants in the above k ( − ) 0 du = p + ∧ a u 2 + F x ( 5 + − 2 e µ + F du F Q dy/dt = dx = = = ≡ µ ) 5 3 3 dy u ,e F F dx H ( 2 A . It is useful to introduce the new radial direction 2 a K are functions to be determined and e p ( 2 x ρ +6 − +3 2 p 2 ρ x e F, f, k = = = Motivated by the form of the deformed conifold metric (3.8) and (3.10) let us In the presence of matter the full system (5.15)–(5.18) may be solved by first y z 2 2 e ds corresponding kinetic and potential terms in (5.7) are found to be similar to (5.11) reproduce (using the ratios of the first“internal” and 5-d the part second, of the and metric the (5.1), second i.e. and the third coefficients in the so that they become are the same asof in the the equations resolved then conifoldconstants) to become metric, the in equivalent system agreement (for in with the the (4.1). special previous section. choice The5.2 of rest Deformed the conifold integration case Let us now complement the discussionstrating of the explicitly deformed that conifold case the in firstpotential [11] order-system which by demon- has there essentially also the follows same from structure a simple as super- (5.14). make the following ansatz for the metric (cf. (5.1)) will appear in the 1-d action (5.7) are thus For a special choice of the integration constants analytic expressions) the resolved conifold metric (2.17) (cf. (5.1)) The resolution parameter 1-st order system. concentrating on the equations thatfor do not involve The ansatz for the p-forms is the same as in [11] (cf. (5.2)–(5.5)) previous case (5.2), (5.3)for and the (5.4), p-forms we are explicitly ensure satisfied that automatically. the The Bianchi independent identities functions of where JHEP11(2000)028 x, y, p (5.25) (5.24) (5.26) ]+ 2 ) F . )] − )+ 2 0 F P k y − (2 2 y e 2 P given in (5.23). The y, e + (2 2 + f 0 F, 2 y f sinh + + 2 y K, F p 2 − f,k,F x y y 4 p − K, 2 2 e 2 , e x kF − − ( 2 p i e . = x 4 Φ+4 [ − 2 + 0 2 p e 0 e x = 0. The equations for sinh 4 2 − 6 1 F Q e p Φ − [ x =0 = 8 K 6 1 2 x Φ − K 0 0 e − x 2 and both reduce to the “standard” e e − 2 1 4 Φ x − Φ + − 2 p f e + + p 4 x − + 4 . In this case the system (5.26) becomes 2 e 2 2 p 0 e ) 2 K, y − y = 2 1 P +2 p Φ 2 x 1 k − ,k , 2 2 16 h 2 e ) + ) = − − Φ f − 4 1 + 1 3 , p e k F x 4 cosh x − 2 f  1 6 e 2 + ( − x 2 − − − 2 2 x − 1 p 2 0 y = − 2 p 2 − f K k, F 2 P p p ( − − x y − 1 2 − p 4 e Φ = f (2 e e x − − y 2 +6 e cosh + e − − Φ+4 + 2 p +2 2 1 p 0 y cosh ,f + 4 − p x 2 y y  p 4 e e − 4 2 1 p − 3 2 1 1 e e 8 p =0 Φ+4 4 e 3 1 cosh − − e − + A, x, y, p, f, k, F, cosh y − 1 8 p 2 e p 4 0 = = = = = 4 e 4 1 + x 0 0 0 0 0 e p x f A F = = b )= 0 )= ϕ ϕ ϕ ( a ( 0 V ϕ W ) is the combination of the independent functions ϕ ( K ab G From (5.10) and (5.25) we find the following set of 1-st order equations for the To show that this system contains the solution of [11] we follow the same strategy Thus there is a“resolved” close and “deformed” similarity cases. (“duality”) between the 1-stindependent order functions systems for the conifold case of [10]. as in the previous subsection: firstto analyze find the that subset the of gravitational ratios sectordeformed of equations conifold the and functions then in include the theimplying metric matter are part. 4 the We + same again 6 as find in the factorization the relation (5.19) case of of the the metric for identical to (5.18)–(5.18) with where and (5.12) The special solution is corresponding superpotential satisfying (5.9)and again is does a not dependstructure sum as on of the the the previous dilaton one gravitational (5.14) and matter parts, i.e. has essentially the same JHEP11(2000)028 = 1 − ) (5.29) (5.27) (5.31) (5.32) (5.28) (5.30) y so that (sinh τ , x 3 2 τ 15 − 2 du , e p q .Asaresult, 4 − e τ e sinh , , − x is exactly the same as K 2 3 ≡− 3 / (5.27) and the following = / − 1 x . 4 +   ) y p x τ 2 1 3 τe +lnsinh + e 2 τ, p − x = 2 dx/dy τ 2(3 y ,dτ + − sinh + . e 2 x τ p 2 sinh b 3 2  2 and τ b sinh 2 = =3 + x, − 2 1 = x y w 3 ,e p τ  − − 6 =tanh 2 1 e − : p 17 − y p − x p 6 6 cosh ) K is the deformation parameter. − τ 3 sinh 2 x − ≡  / , e 4 2 1 and  = 2 τ = 6 1  x ) 2 (sinh 2 y, e x y, q y, e x b 3 2 2 = 2 2 3 − p + p r 8 cosh 6 dτ p sinh + − − we thus reproduce the deformed conifold metric (3.8) K sinh e = e c (3 − 3 x 2 4 / x d b −  4 = = + p = =2cosh  p 1 2 96 3 x 2 1 = e ) 3 e dτ dx p e q dy dτ = = = 6 . It is useful to introduce the new radial coordinate 3 2 y x − − dτ − ) − y, e ,b p x x y 6 e ( − d e =0 ) was defined in (3.9) and c τ (sinh = 0 here lead to the relations which are very similar (“dual”) to (5.20), ( is an integration constant. The remaining equation is 2 − =2coth b K b K x 3 In the general case the system (5.26) can be solved by starting with the equa- It is quite remarkable that making simple ansatze (5.1) or (5.22) for the 6-d Here and in the previous subsection we make a specific choice of the integration constants in − 15 e − Choosing in (5.30). This equation is solved by first introducing where we set the integration constant to zero so that where with dq/dy order to match the standardproduces a metrics more on general class the of conifolds.also metrics. in Keeping The section existence the of 2. integration these constants more general arbitrary metrics was mentioned where combination of the equations for tions that do not involve matter functions: equation for part of the metric(Ricci-flatness) one equations finds admit that aresponding superpotential, the 1-st and 1-d order that action systems the are leadingmetrics the solutions respectively! to resolved of (2.17) the the and associated cor- the deformed Einstein (3.8) conifold JHEP11(2000)028 . , y + x /e B 435 x − ]. p 6 (1998) 253 − The wall of 2 e ; superconformal ]; hep-th/9603167 String theory and Phys. Lett. , =1 , ]. ]; ]. N Adv. Theor. Math. 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