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This content was downloaded from IP address 131.215.225.46 on 12/03/2018 at 22:24 JHEP10(2001)028 D =2SYMin of the deformed October 23, 2001 N 3 S limit of Accepted: N October 16, 2001, Yang-Mills theory from gauged supergravity to ten dimensions in which SU(2) Received: R . The type-IIB solution is obtained by lifting a new solution =2 3 S ısica, Universidad de Buenos Aires ∗ SU(2) T N × L D-branes, Superstring Vacua, Brane Dynamics in Gauge Theories. We find a new solution of type-IIB supergravity which represents a [email protected] [email protected] =7SU(2) =2+1 HYPER VERSION D Ciudad Universitaria, 1428 BuenosE-mail: Aires, Argentina Departamento de F´ Department of Physics, CaliforniaPasadena, Institute CA of 91125, Technology USA E-mail: wrapped branes conifold geometry gauge fields indescribes the a diagonal slice subgroup of are the turned Coulomb on. branch in The the supergravity large- solution Abstract: collection of D5-branes wrapped on the topologically non-trivial Jorge G. Russo three dimensions. Keywords: Jaume Gomis D of JHEP10(2001)028 . 5 R = k N D SU(2) × SYM 2 ) L N gauge fields of D =2SU( N ) Super-Yang-Mills (SYM). truncation of SO(4) gauged N D of the deformed conifold geometry, =2SU( 3 S N 1 =7 D gauge fields for 3+1 SYM 13 D SU(2) gauge fields that solves the full SO(4) gauged supergravity diag ) R SU(2) × L Previous constructions of wrapped branes representing three-dimensional SYM In this paper we find the supergravity solution representing a collection of We find this gravity dual by finding a solution of SO(4) = SU(2) NS5-branes wrapping the supersymmetric 3. The supergravity solution in (SU(2) which admits a metricthe with infrared SU(3) dynamics of holonomy. three-dimensional This supergravity solution describes supergravity is in generalmensions inconsistent. and describes This certain aspects solution of can the then gauge beappeared theory [4, lifted dynamics. 10, to 16, tenin 18](see di- also addition [19]). to the SYM usual theory YM in three term, dimensions a can Chern-Simons include, term in the action with level gauged supergravity. Our solution provides an example of an ansatz with SU(2) gauged supergravity. Supersymmetry requires that we turn on SU(2) 1. Introduction Supersymmetric gauge theories can beliving realized as on the a low collection energy of effectivetwisting field branes which theory wrapping is supersymmetric required cyclesincorporated to [1]. in make The the these topological supergravity gaugenon-trivial realization theories gauge of fields supersymmetric these in [2] wrapped gaugedcan can supergravity. branes then These be by be gauged turning lifted supergravity to on descriptions solutions ten to or the eleven infrared dimensions dynamics and of give rise various gauge a host theories of [3]–[17]. interesting dual Contents 1. Introduction2. Wrapped brane realization of three-dimensional 1 4. Ten-dimensional solution5. Field theory from theA. supergravity description NS5-branes with 11 7 equations of motion, despite the fact that an SU(2) JHEP10(2001)028 3 1 S − =2 N of the 3 S N directions, 3 S in the t’Hooft N − ,thistermarises e 3 S =0). =0inthe k R 3 H with components in the NS5-branes on the R 3 H N There is a curvature singularity in 1 2 , which admits a metric with SU(3) holonomy. In 3 S ∗ T gauge fields, with the aim of describing four-dimensional 1 real scalars of the vector multiplet together with the limit. Finally, in the appendix, we discuss an ansatz for D − truncation can give a solution. In section 3 we find first order N N D SYM ) N The solutions describe a slice of the Coulomb branch of the gauge theory, which is This paper is organized as follows. In section 2 we describe the low energy ef- It is well known that monopoles in this theory generate a non-perturbative su- SU( The real scalar of the vector multiplet arises from dimensionally reducing the four-dimensional =2SYM. 1 = 1 vector multiplet. parametrized by the so they describe 2 + 1 SYM without Chern-Simons term ( scalars obtained upon dualization of the photons. deformed conifold geometry limit, so they are not visible in the supergravityfective approximation. field theory onthe required the SU(2) wrapped branes and study the conditions under which when the background has a non-trivial RR field NS5-branes with SU(2) equations which solve thetion. seven-dimensional gauged In supergravity section equationsfamily 4 of of the solutions mo- solution offor is type-II different supergravity. lifted values We to of discuss tenpectations. the the dimensions, parameter. behavior We giving of Section calculate the a 5theory the solutions makes one-parameter effective in contact action with the on field large- the theory Coulomb ex- branch of the gauge order to obtain atheory globally on supersymmetric the gauge branesof theory to five-branes the one in flat gauge must space, fieldsThen, couple that supersymmetry which the is, can couple the field be field to realized theory by the finding must R-symmetry constant be and currents topologically “covariantly” twisted constant [2]. This gauge theory can be realized by wrapping 2. Wrapped brane realization of three-dimensional directions [10]. The present supergravity solutions have In the supergravity description based on D5-branes wrapping the supergravity solution, related tospace the of NS5-brane sources. thewithin We gauge analyze supergravity theory the the moduli existence byas of sending required a by a complex supersymmetry. moduli probe space into with the a geometry Kahlerperpotential metric, and which destabilizes reproduce thesupergravity Coulomb solutions branch exhibits [20].the a effects On Coulomb that the branch. induce other lifting This of hand, the is our Coulomb not branch are inconsistent, of since order N N JHEP10(2001)028 R × 2) (2.2) (2.1) (2.3) , , under SU(2) spin con- 3 3 × S S L subgroup of the SU(2) 3 . diag × ) 3 R diag S ) , D . , are vector indices of the ) ) =0 SU(2) 2 1 = 2 vector multiplet. SU(2) .  , , i, j ) × 1 1 R-symmetry group, gives rise to a SU(2) × N  , , 3 , , L ) ) , ij R 2 3 × 2) which appears in the near horizon 3 3 2 , , γ and 3 , , , 2 1 3 ij S 2( ( ( 1 1 3 e S ( ( A SU(2) S ⊕ ⊕ ⊕ fill the 4 1 , ⊕ ⊕ ) ) ) ) × ) ) 1 + 2 1 1 =(SU(2) L , 1 1 , , = 2 theory is obtained by identifying the , diag 3 , , ab 2 2 2 1 D 3 1 , , , Γ =(SU(2) -symmetry group of a five-brane appears as a N 1 2 1 ab R , , , ω 1 2 3 does not admit zero modes and therefore an untwisted diag 1 4 3 S + . ∂ , gauge : (  -symmetry group of the five-branes in flat space. scalars : ( spinors : 2( diag R ) = =0 L gauge : ( R scalars : (  1) vector multiplet on the five-branes under the symmetries spinors : ( , ∂ D subgroup of the SU(2) = 2 massless vector multiplet, one decomposes the original SU(2) L SU(2) =(1 N × R-symmetry group of the NS5-brane so that the supersymmetry = 1 theory. In that case the supersymmetry generators are singlets under 3 × S R N N L 32 are tangent space indices on the S SU(2) are spin connection with the SU(2) =(SU(2) × 3 a, b diag S L diag In supergravity, the SO(4) global Turning this background gauge field effectively changes the transformation prop- To verify that the field content that follows from this twist gives rise to a The standard Dirac equation on A different twisting, used previously in [3], in which one identifies the SU(2) 2 3 Then it follows thatSU(2) the transformation properties of these modes under SO(1 gauge symmetry of thereducing seven-dimensional type-II effective supergravity supergravity on theory the obtained transverse by where SO(4)=SU(2) left unbroken by the background, which are SO(1 erties of all fields under a modified tangent space symmetry group of generators are singlets under SU(2) which the unbroken supersymmetriesing transform giving as rise scalars. toSU(2) a The three-dimensional topological twist- SU(2) spinors on Therefore, the modes singlet under SU(2) supersymmetric theory cannot be realized on it. three-dimensional six-dimensional nection with the SU(2) SU(2) four-dimensional JHEP10(2001)028 − (2.7) (2.4) (2.6) (2.5) µν kl F .These a ij µν ij A F 1 A − jl e T 1 − ik with a linear dilaton e T 3 2 e S / 1 . × − gauge fields 12 R Y A D 8 1 , . × 5 ) ) , ≡ − 1 3 3 3 , , li R . 1 3 e T ( ( ij µ ] ⊕ ⊕ R D ) ) transforming in the two-index sym- jk ,A 1 1 , e A, T T ij , , , [ µ 13 ij T 3 1 ) g SU(2) T A D ij , 4 1 + T / × 4 − 1 :( :( kl ij e − L T  ≡− ij ij 1 2 Y dT 2 T − A  ij e ii T = = =det( e T 1 4 ij ij  Y e T − − ,A scalars DT Y and ten scalars ij Therefore, seven-dimensional gauged supergravity is a µ 23 e T ij A 4 ij A e Y∂ gauge fields T ≡ µ 2 ∂ 1  2 A − 2 / gauge fields to be compatible with the full SO(4) equations of 1 Y Y 5 D 16 2 g − gauge fields may in general act as non-linear sources for the gauge 1 2 . We have studied the SO(4) gauged supergravity equations following R D − 34  g ,A . 24 √ R The bosonic lagrangian of SO(4) gauged supergravity in the absence of the two- For the case under study, one needs to turn on SU(2) The bosonic content of the SO(4) gauged supergravity is the graviton, a two- = We recall that the near horizon geometry of a five-brane is ,A 4 14 L The SU(2) By turning on somehaving fields a in space-time this interpretation as gauged wrapped supergravity, branes. one canform construct is solutions given by [21] metric representation of SO(4) = SU(2) Any solution to thedimensional SO(4) type-IIB gauged supergravity supergravity. equationsfields lifts Explicit in to terms formulas a of [22] solution the for of seven-dimensional the ones ten- are ten-dimensional given in section 4. where natural arena to construct supergravity solutionsgauge in field which a is background turned R-symmetry on. solution form, SO(4) gauge fields solution of the five-brane. can be identified as the following components of the SO(4) gauge fields from (2.5) and found sufficientwith conditions that only have SU(2) to be met in order for a solution motion. They can be summarized as follows: fields which have been setA to zero. The reason is that they are sources for the fields along JHEP10(2001)028 D + (3.1) (2.9) (3.2) (2.8)
(2.11) (2.10) µν 3 F 3 µν F . The metric can be divided 3 + ji S reduction of the J × µν 2 − D 2 , F 1 lagrangian = , 2 µν a R ij D gauge fields are turned F J w D a + w subgroup of SO(4) is generated . ) µν 1 r
b. D , ( F 4 x 2 3 5 1 6= a µν − . F a ⊕ requires the scalar matrix to take c ,e + 3 x 2
w D 2 =7 y/ ,e ⊕ , ∧ x − dr b 3 1 x . The SU(2) . D 2 ,e a w + x − J  e e
5
abc 2 4 1  x , 6 1 2 1 − − R e = diag x = 0 for 4 µ − 2 a b , 2 real scalars, the two-form and the metric can y/ a x scalars : 2 F ds 2 x∂ e dw A µ − ) e ∂ = = 2 supersymmetric solution, we will consider seven- ∧∗ r 3 ( gauge fields : 2 a f ij 2 +6 F − N T e x y 2 µ = e 3 and rotation generators 2 7 y∂ a µ ds 2 gauge fields ∂ K 5 y/ 5 e D 16 2 gauge fields must satisfy the following condition g as − 2 1 D D R + are left invariant SU(2) one-forms satisfying  a g w √ , and the most general four-dimensional symmetric matrix invariant under rotations is of the Once these conditions are satisfied one can perform an SU(2) a = This can be easily understood as follows. The SO(4) generators J 5 Only SO(4) matter modes of gauged supergravity that are singlets under SU(2) The SU(2) L where • SO(4) gauged supergravity lagrangian (2.5), giving the SU(2) on. The worldvolume of the five-branes we are interested in is then only SU(2) Any solution of thisSO(4) lagrangian gauged with supergravity gaugedimensional equations, fields type-IIB and satisfying supergravity. (2.10) it fully can solves be the lifted as a solution of ten- ansatz compatible with the worldvolume of the five-branes is the following form dimensional SO(4) gauged supergravity, in which only SU(2) 3. The supergravity solution in In order to construct the can be turned on.under Since SU(2) the non-trivial SO(4) gauged supergravity modes transform • form (2.9). be turned on. Moreover, invariance under SU(2) into boost generators by JHEP10(2001)028 (3.4) (3.5) (3.3) (3.7) (3.6) gauge fields D θdϕ , ,
, x sin 6 ! − β 2 e ˙ f − cos x +6 2 − ˙ . − f e (see below in section 5), it is ˙
a a x 6 βdθ . − +6 . ··· c e b , x +18 2 w ) 2 − e 6= , )+ which the branes are wrapping. , we arrive at the following effective V =sin 3 ). It is easy to show that this ansatz ∧ 2 x  , a , h a 3 2 r b 2 ˙ , 1 a a ( w − 2 − S  +3 w f  e g g R  f x 1 2 T ( 4 5 2 2 2 1 ( 2 abc x 4 e − 6 −  α − a f +6 +6 2 − ds 4 g = e 1 x = 0 for 2 ) e 8 +3˙ h = spin connection with the SU(2) 2 r 3 2 b ( )= e y/ 2 2  = a , 3 − a f ˙ − r 3 a + F f 2 S 3 f 2 e ( A f θdϕ , w 2 e 5 a eff 6 = 2 x y 2 e f 6  e 3˙ L a g ∧∗ 2 − 3 3 sin e

F = a = − a a x 2 3 β f 2 f F 2 α g 1 a 2 parametrize the 2 st 2 2 ˙ − h 2 f e e e h 2 3 ϕ 4 ds 2 1 e − − θdϕ . − − +sin e the various terms in the action (2.11) take the following form = = = = 2 2 and and g 3 α f R 2 2 4 pot βdθ a scal +cos L f − L gauge L 2 = =4˙ β,θ ,etc. L − dβ = e T V y =cos = = 1 3 ’s given in (2.7), and the corresponding curvature is given by dα/dr h a w w 2 e A ≡ α We find the equations of motion by reducing the problem to that of an effective The string metric for wrapped NS5-branes typically has no warp factor in the The identification of the SU(2) is compatible with the equations of motion that followquantum from mechanics for (2.11). the ansatz,sults which in together the with gauged supergravity a equations hamiltoniantogether of constraint motion. with re- Using the ansatz (3.1) and (3.3) natural to look for solutions with We note that thissupergravity equations gauge since field configuration is compatible with the SO(4) gauged that follows from (2.1) requires that Defining lagrangian with the parallel, flat directionsmetric along is of the the brane. form [22], Since the ten-dimensional string frame where ˙ where the angles One can describe them in Euler angle parametrization, where JHEP10(2001)028 , x 2 e (3.8) (3.9) (3.10) (3.12) (3.13) (3.11) (3.14) (3.15) = u = 0 arising V + T , . = ) ) x gdr , ρ ρ − , ( ( H h = 4 4 2 2 , and defining , / / ) − 3 1 x r . ( e dρ W I − − h ) x g I 4 1 h ) 2 . Taking as principal function 0 1 ) 2 − 2 e ρ c } 0 h − ( 2 − c 2 2 2 g e + − 2 / − g 1 ) − e x 3 4 3 ) for the modified Bessel functions 3 g ) is determined by direct integration − W z α, h, x ρ + e x ( { )= + ( ν ∂ 2 g , x )+(1 r I ( x α 3 ( ρ = x )+(1 3 ,u − ( ρ + 1 ρ i − − 12 4 e 7 ( h ρ x / ). For supersymmetric configurations it is ϕ 3 4 2 4 g 3 8 as follows: e ge r / + − → ( 1 − . 2 g e I I 2 + + + 2 ) 0 0 W 1 g − r x u x
c c ,x e 1 ) W = = g − ge − r h = 4 ,where ( 3 u i ∂ ) ( ρ W W ( =3 + = h x =1 ,h x 1 12 ∂ ∂ 2 ) u u e r 6 6 1 1 3 W W ( = ρ 1 4 − ∂F/∂ϕ − α 1 1 4 2 V )= = = = = ρ = i − ˙ ˙ ˙ x h α p u( dρ du dρ dα ) and if the potential of the effective quantum mechanics can be h, x ( W α is a parameter of the solution. Then 2 e 0 c This effective lagrangian together with the constraint = then one can findsolution for first (3.8) order is equations which solve the equations of motion. The from diffeomorphism invariance in theequations radial of coordinate motion determine the for second order F eq. (3.10), (3.11) and (3.12) take the form Thus the first-order (BPS) equations are: and 4. Ten-dimensional solution Since we have fullythe solved corresponding the type-IIB SO(4) supergravity gauged solution supergravity in equations ten dimensions. one can The write expression By a change of radial coordinate The general solution to eq. (3.14) is given by where we are using the standard notation written in terms of a superpotential of (3.13). These functions have a simple behavior, which will be described below. always possible to find athe set equations of of first motion. order differential Thislinear equations can equations whose be are solutions done solve in the present case. The Hamilton-Jacobi JHEP10(2001)028 ,  via
(4.4) (4.5) (4.1) (4.2) (4.3) 2 4 N dµ x 2 e +
2 3 − k Dµ µ j , ). + µ 2 r k 2 2 , ( . φ 4 i
f −
, j T j 4 4 i j Dµ .
. This is in the parametri- µ ψ, cos 3 j µ − j i 2 1 α 4 3 ik + ψ. Dµ i i i φ T 2 1 T Dµ DT cos 3 i )= jk i Dµ , x r sin Dµ ∧ 4 Dµ ( =1)and 1 ii gA ∧ 2 − ψ =cos y Dµ i i − Dµ T ij e 1 2 x 4 i µ + 2 T − i ∧ ∆ ij 1 − + Dµ µ 2 , T − kj e i ( − 4 Dµ j 1 ψ =sin T ∧ / i ∆ j 3 µ 3 2 2 ∧ 1 1 2 ik G F µ i e S − ij i − ,µ 1 8 i , g µ ∆ 2 + gA ∆ 2 j gA Dµ φ g a jk + − µ 3 + 3 i T w + g 2 7 a µ ,µ i ij ik sin ∆=sin − − UDµ 4 1 + ij ds 1 T w φ dµ dT T 2 7 φ 2 y/ ρ =1) 4 i 2 − 0 3 / ds i x = = =2 1 + α 4 2 cos sin − i i / 2 Y 1 ij e 1 U i ∆= ψ ψ  dρ ∆ = 2 ≡ Dµ , x DT 2 2 2 st 8 ∆ e G / 1 / , 2 . Using (4.3), (3.1) and (2.9) the metric in the string frame 3  1 =sin =sin ∆ g ds 1 6 Y N − Y 1 3 N µ µ G = = = = x + − 2 φ
2 E α 2 2 H , 2 e /g 1 ds − given by ’s parametrise the transverse i R 3 ρe µ 0 e S 2 φ 2 arises as the integration constant in eq. (3.13) for e ds 0 part, with the identification we made earlier = = φ Let us now write explicitly this new type-IIB supergravity solution. First, we 2 , φ 2 st 1 2 e ds where and where the The metric in the string frame is therefore Using (2.9) and (3.1),R we note that the string metric has no warped factor in the of the ten-dimensional Einstein framefield metric, strength the are dilaton given and in the terms NS-NS of three-form the gauged supergravity fields by note that the seven-dimensional gauge coupling is related to the brane charge andthedilatonaregivenby( zation of the relation 1 JHEP10(2001)028 is as 0 ρ depends , 0 = integration 3 ρ µ 0 ρ 1 c w which results in a . Therefore, the 2 1 , 0 2 , /
− 3
ψ − 1 x 3 ρ µ 0. The figure shows the 5 − 3 ρ<ρ ∼ w cos 2 < ,e φ x 2 1 2 2 0 ρ, c ,e 1) + x ,e log ˜ 2 4 − 3 ,e 4 3 ρ x 2. x 0for dµ . 4 is real). The value of e 1 ∼ = e ∼ = > = . The behavior near x , α 0 2 − 4 2 Nρ. +( . ρ 0 3 0 x diag 8 , 4 =2 e 0 → ρ, e y/ h ,α , e − 2 0 2 3 9
5 c . becomes negative at x 2 = 4 ∼ = 0 2 ˜ . ,Dµ ρ 1 e x 2 3 − ij − 4 2 µ µ , u − O 1 2 ,e 6 e − truncation (2.7), one has the following covariant w w ) = + − 1 2 2 1 1 α D 1 x ρ 2 = 2 vanishes at some 2 y + + − 0 ) =˜ + − c ρ e 2 1 x ( dρ , T (since the scalar field x 6 µ µ h x =0that 2 in figure 1). There are three different regimes: 3 2 ij 2 0 − − ρ e (3 ρ − x w w dρ e 2 e 2 5 dT 1 4 1 2 2 1 1 − = 0.4 0.6 0.2 0.8 exp -2x e = = = − − ρ 1 3 y ij U dµ dµ dT given in (3.4) and = = 0, the function 1 3 ij The function F < = 0 we have near Dµ Dµ 0 0 6 c c and it decreases towards zero as 0 Let us now examine the behavior of the solution as a function of the For c For 6 space terminates at derivatives on different curves for the values follows: where we have used thetopological gauge twisting. fields in One thecurvature can ansatz (3.3) also that write implement down the the required three-form flux using (4.1), the constant (see plot of Figure 1: Taking into account the SU(2) a) singular solution, which is an acceptable one according to the criterium proposed by [1]. JHEP10(2001)028 0 φ c − e 0is (i.e. , 0as (4.6) (4.7) (4.8) − )and
.The .The ρ 2 < 2 2 ). The 4 ) → e S = 1 0 ∞ / φ c 2 dψ ρ / × E . 00 1 in − + 1 ) g =0andwe 0  e ρ S
ρ 1+1 2 4 ρ 2 2 ρ (( ∼ = goes to − dµ dφ O 1 The metric with this x φ ρ + 2 φ ( e e ρ. 2 3 )+ 2 − 1 . ∼ ρ = 4 Dµ = ∼ / α 3 2 − u +sin ρ + E ρ 00 2 1 ,e 2 2 g diverging at 2 ∼ / 3 dφ =( α E 00 ρ Dµ 2 x g degenerates to an 2 2 ∼ + ρ e 3 φ 2 1 , = 2 3 + 1 SYM. The parameter e S 2 1 +˜ ,givenby 0 ρ 2 ,e N c Dµ ˜ ρ 1 in [14, 24, 25] ρ, e d − . There the dilaton 0. For a generic value of the integration + ) γ 0 ρ ∼ a ρ ˜ ρ (2 → N w φ 10 7 2 a = ρ ∼ = and ) SYM. Below we discuss this correspondence + w ρ x , so the singularity is again a bad one and we a k N 2 2 ρ as =0: − is, for any w e ρ a + . This is in fact the same leading behavior as the ∞ ,e 2 w ∞ ρ →∞ 2 0 / . dρ 1 φ 2 = blows up at some 0 1 4 =0is − log − ˜ =2SU( ρ Nρ ρ ρ x e ρ ρ 1 2  ) 2 N + − ψ ∼ + e N (
goes to + x 2 2 ρ 2 , + x 1 − 2 sin
= e R − 2 0 , ). Thus e 1 φ 0, we have near 1 2 2 ρ . The metric and dilaton near the singularity are then given by R 1, − 0 > ds e ρ ρ, α − 0 2 ≤ = 0 is deemed to be unacceptable as a dual description of a field theory [1]. ρ = = ∼ ∼ − 0 ( ρ ds 1, the factor . >c ρ ∼− = > NS5

f 2 +6 θdr 2 ˙ x 3 − ) x dθ , a a 2 − − a 2 a x e e r 14 , one can express that term in terms of first 2 2 2 y/ − (  ∧ ˙ e a e f − +3˙ A = = A 3 )sin x ( 2 dr 2 r h =0, α +2 2 ,sowealsohave ˙ ) ( y 2 − θ 2 2 +6 r ˙ r b e e y/ f e 2 A +12 ( x y/ 5 2 2 F − 16 ˙ a 2 ˙ e e a A − x f e 2 2 f 2  ˙ 3 a a g 3 − ∧∗ 2 =sin = = 2 e

f a a a 5 2 2 2 3 1 2 f f e F g 1 a 3 y/ 5 2 following from (A.4). It contains a term having second +16 2 f e e + gF gF gF g 3 1 f 2 2 2 1 e − − 2 5 f ˙ a a being interpreted as time. One must also impose the zero e 2 2 = = = = + r g  f 2 R 5 f a pot e scal 3 . These gauge fields preserve the SU(2) symmetry of the L 1 2 2 e L gauge L , and defining 2 L f a − − . Using the equation for 4 a = − dA/dr = H . This is consistent with the equations of motion. Indeed, consider the and using the hamiltonian constraint = f y ˙ f 4 A 4 0= − − One way of solving the equations of motion is to insert the ansatz into the = = y derivatives. The resulting equation is the same as the equation for derivatives of to depend on the radial coordinate lagrangian (2.11) and solvechanical the model equations with of motion of the effective quantum me- equation of motion for where have the property that theysatisfy are the pure condition gauge (2.10) when Setting energy condition on thethe system. following contributions Plugging into the lagrangian our ansatz one gets Since we are interestedof in section NS 3) five-brane toy solutions, look for it solutions is with natural no (as warp in factor the in solution the parallel directions, i.e. which has the following field strengths The ten-dimensional string frame metric is of the form JHEP10(2001)028 , × 04 Nucl. (A.7) (A.8) (A.6) B 613 , ]. SU(2) ]. (2001) 025 super Yang- super Yang- . 06
=1 =1 x 6 − N N Nucl. Phys. e , − x hep-th/0105049 2 J. High Energy Phys. − 2 hep-th/0103167 e ˙ ]. ]. , Supergravity duals of gauge A x ]. ]. 2 ]. +6 − limit of pure limit of pure x h 2 2 e N N − , 3 e (2001) 024 [ ) 2 J. High Energy Phys. 1 2 g V (2001) 028 [ g 09 , Fivebranes wrapped on associative three- 2 1 − − M-fivebranes wrapped on supersymmetric 04 ]. 2 T − x ( hep-th/0011190 hep-th/0012195 D-branes and topological field theories 2 3˙ α 15 hep-th/0008001 hep-th/0008001 2 hep-th/0011288 1) e − 2 − = ˙ Massless 3-branes in M-theory h 2 2 Towards the large- Towards the large- eff The supergravity dual of a theory with dynamical su- A Supergravity duals of gauge field theories from D6 branes and M-theory geometrical transitions from ( − L x ]. 2 2 nez, nez, α − (2001) 588 [ (2001) 588 [ nez, h hep-th/9511222 (2001) 008 [ 4 (2001) 106003 [ (2001) 126001 [ − =4˙ 86 86 J. High Energy Phys. Supergravity and D-branes wrapping special lagrangian cycles e 03 , 2 T J. High Energy Phys. g 1 , 2 D63 D63 gauged supergravity in six dimensions ]. + 4 uandC.N.Pope, F h (1996) 420 [ 2 − hep-th/0105096 e 2 c, H. L¨ − Phys. Rev. Phys. Rev. nez, I.Y. Park, M. Schvellinger and T.A. Tran, Phys. Rev. Lett. Phys. Rev. Lett. B 463 , , gauged supergravity in five dimensions = u˜ , , V hep-th/0105019 (2001) 025. gauged supergravity Mills U(1) persymmetry breaking [ Phys. (2001) 167 [ Mills cycles J. High Energy Phys. cycles theories from [1] J.M. Maldacena and C. Nu˜ [8] J.D. Edelstein and C. Nu˜ [9] M. Schvellinger and T.A. Tran, [2] M. Bershadsky, C. Vafa and V. Sadov, [3] J.M. Maldacena and C. Nu˜ [4] B.S. Acharya, J.P. Gauntlett and N. Kim, [5] H. Nieder and Y. Oz, [6] J.P. Gauntlett, N. Kim and D. Waldram, [7] C. N´ References one obtains the following lagrangian and with It would be interesting to search for BPS equations using this effective lagrangian. [10] J. Maldacena and H. Nastase, [11] M. Cvetiˇ JHEP10(2001)028 , , , ]. . =2 Nucl. Phys. N , , (2001) 3 ]. . (1998) 046004 Supergravity and hep-th/0008076 hep-th/0108080 B 606 D58 ]. , gauge theory/gravity ]. =2 hep-th/0106117 N ]. hep-th/0106055 Wrapped fivebranes and Phys. Rev. , supersymmetric membrane flow , Nucl. Phys. (2001) 044009 [ gauge theories from wrapped five- Gauged maximally extended super- , Membranes wrapped on holomorphic (1984) 103. hep-th/0008081 =2 =2 hep-th/9911161 D63 N N Gauge theory and the excision of repulson ]. (2001) 106008 [ ]. Gauge dual and noncommutative extension An B 143 The enhancon and (1982) 413. Instantons and (super-)symmetry breaking in 16 hep-th/0106160 D64 (2000) 022 [ Consistent Kaluza-Klein sphere reductions Phys. Rev. supersymmetric RG flows and the IIB dilaton , 10 B 206 M-fivebranes wrapped on supersymmetric cycles, 2 (2000) 086001 [ =2 Phys. Lett. , (2001) 269 [ hep-th/0003286 N D6 branes wrapping kaehler four-cycles hep-th/0004063 Phys. Rev. D61 . , B 519 Nucl. Phys. , ]. ]. . . . Curvature singularities: the good, the bad and the naked uandC.N.Pope, Branes wrapped on coassociative cycles (2001) 209 [ Phys. Rev. supergravity solution D-branes, holonomy and M-theory , limit of theories with sixteen supercharges (2000) 064028 [ J. High Energy Phys. N , c, H. L¨ =2 andez, Phys. Lett. hep-th/0105250 , , B 594 N D62 hep-th/0103115 hep-th/9802042 curves super Yang-Mills theory branes hep-th/0109039 gravity in seven-dimensions (2+1) dimensions [ hep-th/0107220 Rev. Phys. hep-th/0002160 of an the large- [ singularities RG flows [13] R. Hern´ [12] J.P. Gauntlett, N. Kim, S. Pakis and D. Waldram, [14] J.P. Gauntlett, N. Kim, D. Martelli and D. Waldram, [15] F. Bigazzi, A.L. Cotrone and A. Zaffaroni, [16] J. Gomis and T. Mateos, [18] J. Gomis, [17] J.P. Gauntlett and N. Kim, [21] M. Pernici, K. Pilch and P. van Nieuwenhuizen, [19] R. Corrado, K. Pilch and N.P. Warner, [20] I. Affleck, J. Harvey and E. Witten, [22] M. Cvetiˇ [24] K. Pilch and N.P. Warner, [23] S.S. Gubser, [25] A. Buchel, A.W. Peet and J. Polchinski, [27] C.V. Johnson, A.W. Peet and J. Polchinski, [26] N. Itzhaki, J.M. Maldacena, J. Sonnenschein and S. Yankielowicz, [28] N. Evans, C.V. Johnson and M. Petrini, JHEP10(2001)028 , ]. AdS/CFT RG flows hep-th/0107261 =1 N ]. ahler structure of supersymmetric (2001) 014 [ 10 The K¨ Probing some supersymmetric Coulomb flow in IIB super- 17 hep-th/0011166 =1 N An (2001) 036 [ . J. High Energy Phys. 05 , hep-th/0106032 , gravity J. High Energy Phys. holographic RG flows [29] A. Khavaev and N.P. Warner, [30] C.V. Johnson, K.J. Lovis and D.C. Page, [31] C.V. Johnson, K.J. Lovis and D.C. Page,