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This content was downloaded from IP address 128.194.86.35 on 02/10/2018 at 21:13 JHEP06(2001)025 . June 11, 2001 . Two other spatial [hep-th/0101202] 3 S Accepted: . We also show how to obtain that 3 May 4, 2001, S M University & Received: gauged supergravity in five = 4 supergravity theory obtained in U(1) N × Dynamics in Gauge Theories, AdS-CFT Correspondance. We study the SO(4)-symmetric solution of the five-dimensional [email protected] [email protected] SU(2) U(1) gauged × HYPER VERSION Center for Theoretical Physics, Texas A Center for Theoretical Physics Laboratory for Nuclear Science andMassachusetts Department Institute of of Physics Technology Cambridge, Massachusetts 02139, USA E-mail: College Station, Texas 77843,E-mail: USA solution from six-dimensional Romans’tion theory on to a massless circle. type-IIAis We supergravity. defined then on The up-lift the the dual worldvolume solu- gauge of field a theory NS-fivebrane is wrapped twisted on and SU(2) Abstract: Tuan A. Tran Martin Schvellinger Supergravity duals of gaugefrom field theories This solution contains purely magneticbe non-abelian interpreted and electric as abelian a fields.which reduction comes It of can from seven-dimensional type-IIB gauged supergravity supergravity on on a torus, directions of the NS-fivebranethree-dimensional are gauge field on theory a with torus. two supercharges. In theKeywords: IR limit it corresponds to a dimensions JHEP06(2001)025 regions, 3 . In particu- 2 AdS T × and 3 4 S [12], keeping the radii 2 AdS ahler 4-cycles, special la- T × 3 S holonomy manifold. We consider 2 G -type regions to 6 1 SYM theory on a torus 7 AdS =1 N limit of D4-branes on 2- and 3-cycles, as well as, wrapped NS-fivebranes wrapped on N N is embedded in a seven-dimensional 3 S In this paper, we concentrate on a system which, when up-lifted to ten dimen- When brane worldvolumes are wrapped on different compact spaces [4]–[10], 5. Discussion 13 lar, this fixed. Since the braneKilling worldvolume is , curved, the in resulting orderIn field to order theory define covariantly to on constant the describethe brane the NS-fivebrane worldvolume flows will worldvolume between be in the twisted. the 5+1-dimensional UV) field and theory (defined the in 2+1-dimensional field theory in by considering the large- NS-fivebranes [9]. sions, can be interpreted as type-IIB NS-fivebranes wrapped on 1. Introduction Finding dual field theoriesmanifolds of gives gauged the supergravities possibility withlated solutions to to involving explore twisted curved some fieldFurthermore, aspects this theories of searching through certainly supergravity the provides theoriesity new well-known re- which examples turn AdS/CFT of out duality the to AdS/CFT [1, be dual- 2, interesting on 3]. theirthere own are right. several situationslarly, where fivebranes twisted and gauge D3-branes field wrappedAlso, theories on [11] fivebranes holomorphic appear. [6] curvesinvestigated, Particu- and were while studied D3-branes extensions [4, [7] to 5]. grangian wrapped M-fivebranes 3-, wrapping on K¨ 4- associative andsystematically 3-cycles 5-cycles, studied have co-associative in been reference 4-cycles [8].ity and solutions In Cayley describing a 4-cycles recent the have paper, flows from been we have studied supergrav- Contents 1. Introduction2. The Romans’ theories in3. 5 dimensions Obtaining five-dimensional Romans’ theory4. Dualsof3-dimensional 4 3 1 the decoupling limit of JHEP06(2001)025 , × =4 bad =4 N N . SU(2) 3 × S 2) × , 1 S . However, it was 1 /g 2) AdS group in five , U(1) generically leads , whereas the other two × 3 S = 0, i.e. the U(1) ungauged -like manifold since its spatial 1 . The dual twisted field theory 4 g 3 the singularity. It would be inter- S , with two supercharges. It is worth AdS 2 and T 2 g × 1 remove = R 2 g is always assumed to be non-zero because in the , corresponding to U(1) and SU(2), respectively. 2 1 g 0 can be taken after some appropriate re-scalings g (4) gauged supergravity theory [15]. Several aspects 2) group is isomorphic to SO(4 = 4 supergravity constructed by Romans [14]. That → F , and 1 1 g N g 2) whose maximal bosonic is SU(2 | 2 , . In the Romans’ paper 1 g If we turn off the electric abelian fields, it is also possible to find a solution for This paper is organized as follows. In section 2 we review the basic formalism and and 2 to two coupling constants dimensions. The fieldin content and the the next lagrangiansupergravity section, of given this however in theory here [14]. will we In be briefly fact, introduced discuss the gauge some group features SU(2) of the kinetic term for the self-dual tensor it enters as the factor like 1 pointed out that the limit is defined on the NS-fivebrane worldvolume wrapped on noting that this theory does not come from an In the theories analyzed ing [14], four cases are considered depending on the values of spatial directions are wrappeddimensional on twisted a gauge torus. field theory In on the IR limit it corresponds to a three- the five-dimensional gauged supergravitycriterion [13], given in which reference indeed [4],so one is can that singular. see in that Using the theone singularity the IR of may that say this solution that solution is theesting electric does to abelian not know fields represent whether the ais non-singular gauge related solution to field with the theory. non-vanishingbe rotation abelian Therefore, of related 2-form the to the NS-fivebrane. mechanismby If studied rotation. it in were reference the [16], case, leading it to would a probably desingularization set-up of the five-dimensionalAdS supergroup Romans’ is theories SU(2 [14].U(1). The Furthermore, five-dimensional the SU(2 the IR, we will startdimensional with the SU(2) SO(4)-symmetric gauged solution, obtained by [13] of the five- solution only contains magnetic non-abelianhow and the electric five-dimensional abelian supergravity fields. can bedimensional We will obtained gauged show from supergravity a reduction theory ofis on the obtained a seven- by torus. reducing This type-IIB seven-dimensional supergravity theory on directions does not live inthe the torus spatial in sector the oftheory ten-dimensional five-dimensional is theory. supergravity, but confining. In on addition,as On we a the will reduction other see of hand, thatthe six-dimensional the in Romans’ gauged five-dimensional the six-dimensional supergravity theory on IR canof a the this be circle. theory viewed This as relates seentwisted it from gauge the to field gauge theories, field havepreviously theory been mentioned point analyzed solution of in to view, reference massless including [9]. some type-IIA We dual then supergravity up-lift on the and dualization [17, 13].symmetry. Indeed, we Hereafter, will we study will a assume solution having only SU(2) gauge JHEP06(2001)025 , φ (2.1) (2.2) (2.4) (2.6) (2.5) (2.3) µνρστ ,two ε µ . A , + ab ) b I  µν , , on a torus.  3 (Γ b A  S = 4 supergrav- ) I µν µν νρ ab F A N h φ φ µν ab 6 2 6 2 h 1 8 √ and non-abelian field √ 2 √ e − √ e µν + , ≡ 4 1 , A − K ν b ab µν  A − νρ ab b J µ a I H µν ab µν U(1) gauged ) , A F  ,H H 45 τ × ( ) I , an U(1) gauge potential ρ A µν IJK µν I µ Iµν γ (Γ F I γ ν ρσ µ A F I µ g δ 6 F ab φ 4 1 , 6 √ + Ω I = 4 supergravity from seven-dimensional 4 is the SU(2) coupling constant and µν gA √ 2 µ − φ I µ F e g 6 2 1 3 4 A N A . We are interested in the case in which the − √ νρ µ 4 1 ν ν µ b γ + − ∂ ∂  ( χ e a − 2 µνρστ  ab − − ) ε ≡ 1 acting on the Killing is T µ √ ν φ I ν 1 3 6 µ µ ∇ = 1 SYM theory on a torus is studied in section 4. − A A which transform as a doublet of U(1), a scalar ab µν ∂ √ e µ µ D − = ∂ ∂ are defined as follows 4 1 )( N α µν + b a  φ a B  U(1) gauged ,h ≡ ≡ µ −  ab µ ) ∂ × φ ab T I ( µν µν 6 ) µν ab φ D µ 4 µ √ 1 2 h F F γ 45 ∂ − ( e + + (Γ µ 2 a φ γ and four gauginos and R g  6 2 µ 2 4 1 √ 8 1 1 µi √ D , three SU(2) gauge potentials − ψ + − α µ e µν ab e = = are given by g = H a 2 I µa µν L 1 , √ F δχ 1 6 is the determinant of the vielbein, δψ ab − T e e ≡ ab T supergravity, which is obtained by reducing type-IIB supergravity on where strength respectively. The transformations for the gauginos and gravitinos are where The gauge-covariant derivative 2-index tensor gauge fields We also consider the relationKaluza-Klein to the reduction massless on six-dimensional a supergravitytheory theory circle. and via the The 3-dimensional This flow flow between is the driven 5+1-dimensionaldimensions by gauge obtained the in SO(4)-symmetric [13]. solution Discussion of will the be Romans’ given theory in section in 5. five is a Levi-Civita tensor density. The abelian field strength ity constructed by Romansof [14], a whose conventions we follow. The theory consists In this section we review the five-dimensional SU(2) 2. The Romans’ theories in 5 dimensions theory and there are notthe two five-dimensional 2-index SU(2) potentials. In section 3, we show how to obtain U(1) coupling constant andlagrangian of two the 2-index theory without tensor the gauge two 2-form fields potentials and are U(1) gauge zero. coupling is The bosonic four gravitinos JHEP06(2001)025 (2.8) (3.1) (2.9) (2.7) (2.11) (2.12) (2.10) , =0. I µν , F . For our (2) 2 + =4super- φ 5 . Following 6 2 G 4 S √ Iµν µν T N − = F − F e φ × 2 6  (1) 2 4 µν g 3 √ ρσ F S G e µν φ . 6 g F . 6 1 1 6 4 5 √ 3 √ ρσ , ˆ e F + F − 4 1 ∧ ··· µν  U(1) gauged , , 3 µν g − ˆ a 6 1 F F are the five-dimensional Dirac  × Iρσ ˆ ∗ I µν . =1 µν ··· ˆ − φ F  τ F − F I αβ ρσ F νρ e αβ φ is deformed to are tangent space (or flat) indices, γ F 6 γ F Iµν 2 1 8 I 5 τκ √ ρ µν F S F µ − αβ g µ − ,n e φ α, β ] F 6 1 I ω ˆ 6 ρσ φ 6 n 2 √  2 d α F 4 1 4 − √ . φ − 2 ∧ 6 e . , + 8 I νρ √ + ˆ T φ 4 1 µ I F ... γ στ − στ φ d µνρστ ∂ 6 × ˆ ∗ ε 1 F F 4 − Iρ 2e √  µ α 3 1 1 2 [ I I ) νρ νρ − F − S γ φ − e ≡ F F e − ! µ  2 . In this limit φ a l 1 ∂ g 8 1 5 1 φ n ν  6 ˆ ∗ 6 µ S 4 )( √ − ˆ µνρστ µνρστ 1 = φ∂ R √ φ ∇ µ φ 2 µ 2e n eε eε 6 ∂ = 2 ∂ 4 2 1 1 √ − − ( ...α e 10 1 1 2 2 = =2 = = α L g γ − φ   µν  νµ R +4 R Iνµ F = φ F 6 φ 8 is the spin connection. Indices L 6 √ 4 √ 1 − are spacetime (or curved) indices. The e e − αβ e  µ  ν ω ν µ, ν D D Let us begin with a subset of the bosonic sector of the ten-dimensional type-IIB while The equations of motion of the lagrangian (2.1) are this observation, instead of takingpresented the in [18], singular we limit will of showcan the that be metric the derived SU(2) and gauged from field supergravity type strength in IIB five dimensions on where purpose, we are interested inout getting the the U(1) five-dimensional gauge gauged coupling.as supergravity Turning with- taking off a the singular U(1) limit gauge coupling of can be thought of gravity can be obtained from reduction of type-IIB supergravity on 3. Obtaining five-dimensional Romans’ theory It was shown in [18] that the five-dimensional SU(2) with matrices, After some appropriate rescalingsas of the fields, the lagrangian (2.1) can be written supergravity The eq. (2.12) is the lagrangian presented in reference [17] with JHEP06(2001)025 . and l (3.2) (3.4) (3.5) (3.6) (3.3) 1 3 ˜ ∗ ˜ φ S atze for 10 2 atze (3.3) √ . e . 2  µν g  ˜ g − ρστ ˜ i φ 2 10 8 ρστ ˜ ˜ F F 10 2 ˆ .Theans¨ √ F √ 3 , ∧ , e ρστ 2 S − i i 2 ˜ 2 ρστ ) F i ˜ ˜ h 2 ˆ i 1 g F F . ˜ F ˜ ∗ 2 3 µν ∧ 4 ˜ ˜ µν ˜ g φ i gA 2 F ∧ ˆ g − 9 1 ˜ 10 i 2 F ˜ ∗ 2 − √  ˜ 1 ˜ φ F 12 g i − 1 − ˜ ∗ , 4 ρσ 10 σ e 4 ˜ − ( φ ˜ √ ρσ F . Substituting the ans¨ + e ν 10 k 1 2 10 ˜ ρσ 3 ρσ 1 k =1 ˜ √ F − i A ˜ ν X F √ ˜ h )=0 − ˆ 3 F e ˜ ∧ φ = µρσ ˜ ∧ µν F − ˜ j 1 3 F ˜ g 10 j ˜ ∗ 5 3 µρσ ˜ ˜ 10  1 √ F A ˜ 9 1 h ˜ ˜ ˆ φ F F √ ˜ − φ 10  ∧ ∧ 4 5 e , − , ∧ √ or ˆ i 10 φ 4 i 1 , 3 i 2 − ρ 2 3 √ ˜ h ˜ − − ˆ l , g ˜ ˜ 1 A ν F 1 4 F F e − 3 ˜ 4 ˜ ˜ ∗ (e F ˜ ∗ F ijk e 1 2 ˜ ˜ ∧ ijk φ F ∧  ˜ φ µρ 1 4 3 + ∧ , 3 10 ˜ ∗ 2 2 + ˜ ˆ 10 g ˜ i F 4 2 7 F √ 4 ˜ g F φ ,d √ 1 ˜ ˜ ˆ + s ˜  ˆ ∗ 1 φ e i ˜ ∗ A F 24 2 10 d ν 4 ˆ − ˜ 24 2 φ ˜ ∗ ˜ ˜ ˜ φ φ φ ˜ g √ F φ, e ˜ , , φ g ν − ˜ φ − ˆ i i 10 10 − φ∂ 2 4 2 e 2 − ∧ − 10 i 1 √ √ ˜ 3 10 ˜ µ 10 4 ˜ 10 10 F F φ∂ i 3 i 2 1 8 2 √ ˜ 2 √ ∂ , − − A ˜ µ ˜ √ σ F =e e √ √ e F e ∧ ∧ 1 2 − ∂ ∧ i 4 4 2 2 2 4 2 1 1 1 1 − − = = =e = ˜ i ˜ ˜ 2 10 F = F F 2 i ˜ 3 ˆ F φ = = )=0 )= 2 10 4 2 √ 1 1 ˆ ˜ h F 3 ˆ 1 4 µν ˆ s φ )= )= 3 ˆ i ˆ F d d 2 ˜ ˜ µν R φ F ˜ ˆ ∗ + ˆ ∗ ˜ F d ˆ R ( φ )= )= )= d 2 ˜ ∗ ˜ ∗ i 4 2 ˜ d − φ ( ˜ ˜ ˜ ˜ φ A F F d d (e d ˜ ∗ ˜ ∗ 10 ˜ ∗ 2 d ( ˜ √ ˜ φ φ d = − 10 10 4 2 3 √ √ (e ˜ F − (e D d (e D Following the procedure in [19], we reduce the ten-dimensional theory on into eq. (3.2), we obtain Using odd-dimensional dualization [20] we change 3-form to 4-form In terms of the 4-form field strength, the eqs. (3.4) become retain only SU(2) of a full SO(4) of where The equations of motion of the ten-dimensional theory are the metric, the scalar and the three form field are JHEP06(2001)025 (3.8) (3.7) (3.9) (3.10) (3.12) (3.11) , + 2 2 + ) i i 1 2 ˜ F ∧F gA 2 ∧ . i . 2 − k 1 , k 1 ˜ F i i ˜ ∗F A A ˜ ∗ σ is h ( φ ˜ φ ∧ ∧ 2 6 ∧ 4 , 10 j 1 3 j 1 √ 2 =1 i T 2 i √ ) ˜ X e A A 2 F − 2 1 φ e 6 ijk ijk 3 dZ 3  √ 2 1  =1 − i X − i 2 1 + 2 1 2 e − 4 2 1 F 2 4 + + ˜ g 1 ∧ F i 1 + i 1 4 dY i 2 ˜ k ( A ∧ following [24, 25, 17]. The ansatz for , F = 4 supergravity with vanishing U(1) dA h d φ 4 , 3 2 6 ˜ ∗ . )+ l F 6 ∧ ˜ [23]. Having obtained the seven-dimen- 1 √ = = 2 A T N 5 i 1 ˜ ∗ j φ i i 3 ∗ 2 2 ˜ 6 − φ ∧ h 6 2 ˜ S dZ F φ √ F i 10 2 gA 4 dY , 6 ∧ − 2 ˜ √ √ + F +e i e ∧ e − e 2 2 5 h i ∧ 1 2 2 1 2 i σ 2 g ds dZ ijk ˜ dY − F −  φ and the signature of the space time is mostly plus. ( = and and 6 ∧ . 2 ˜ 1 4 φ φ 4 i √ +4 i φ 2 dφ 2 6 g 5 d h 1 3 √ 1 F F 5 ∧ 24 ∧ l+ 1 , , =e = = ˜ ˜ ∗ φ − 1 dφ 3 ∧A +e i d ˜ 2 7 3 2 φ and ˜ 2 i ˜ ˜ ∗ ∗ ˜ A 2 5 A 2 F F 10 2 ds d F 1 d 2 √ 1 2 ds ∗F e ∧ = φ = − 2 φ, φ − 6 i 2 4 = 2 supergravity in seven dimensions without topological mass l 6 g 2 l 1 6 √ 13 4 ˜ 1 F 5 √ F ˜ ∗ F √ ∗ ˜ 1 4 N +4 R + R = =e =e = 3 ˆ φ 2 10 = 7 ˆ F ˆ s 5 L d L The complete reduction ansatz from 10 to 5 dimensions is The above seven-dimensional gauged supergravityalso whose be lagrangian obtained is from eq.from (3.7) an reducing can appropriate type-IIA truncation supergravity on of a gauged supergravity derived where sional gauged supergravity, we reduce it on where The ansatz (3.12) tellsgravity us can that be any up-lifted solution to of ten dimensions the and, five-dimensional gauged this is super- a solution corresponding to The lagrangian (3.10) is thecoupling SU(2) constant and gauged without twoof 2-form canonical potentials. normalized Eq. scalar (3.10) is written in terms The five-dimensional lagrangian obtained from this process is term [21, 22] Eqs. (3.6) together withdown Einstein’s in equations (3.6)) (for constituteSU(2) simplicity, we the gauged do equations not of write motion them derived from the lagrangian of reduction of the seven-dimensional gauged supergravity on JHEP06(2001)025 , − l 1 2 (3.14) (3.13) ¯ ¯ ∗ , A ¯ 2 φ 2 ∧ 1 √ i dZ 2 =1SYM e ¯ φ F . Therefore, in the same 2 6 1 g 9 ∧ √ 3 N is S 8 i 2 2 S 1 − 4 ¯ F × √ S 3 1 4 is presented above. dY , +e + S × 1 2 i ∧ 2 3 S ¯ l+ F 1 S SO(4). The first corre-  × ¯ ∗ dY i ∧ 3 ¯ × φ φ × h i 2 2 6 S 1 1 ¯ √ √ 15 F ∧ 5) S 8 e ¯ ∗ i , 2 2 F g 3 +e =1 g i X 1 2 4  ¯ +4 φ i 1 2 i 2 + 1 √ ¯ . Reducing the above six-dimensional F k gA − k 1 SYM theory on a torus h e ∧ ¯ A − i ∧ 2 2 i ¯ ∧ j 7 F 1 √ σ type-IIB NS-fivebranes. If the fivebranes h j 1 ¯ ∗ =1 2 ¯ ¯ φ A ∧ 2 N 1 + i 3 √ N =1 is equivalent to type IIB on ijk i 3 h X [9], together with the dualization of the 3-form −  ¯ 3 F e 1 2 φ , 3 ijk S 2 1 6 , 3 ∧  S 9 √ i + ¯ 2 2 8 F 3 × − ¯ i 1 ¯ g F × F − 1 1 ¯ 2 ∧ e A ¯ ∗ 1 ) S ∧ , 24 3 d ¯ ¯ 2 φ i φ i S ¯ 2 2 2 F g 1 ¯ ∂ ¯ ¯ − φ F F = √ 4 ( 2 2 e i 1 2 2 ∧ √ 2 ¯ + 3 e F =1 3 1 i X 2 5 − ¯ 1 2 √ ∗F F ¯ 4 2 1 1 ds φ − − R φ. 6 and φ 4 √ 6 6 = 2 7 e √ 3 ¯ √ 8 )= )= )= 6 A i 3 2 4 ¯  φ d L ¯ ¯ F F is dual to a theory obtained from reducing type IIB on d ¯ ∗ ¯ ∗ ¯ ∗ = =e = = 3 ¯ ¯ φ φ ¯ φ 3 4 ˆ 2 S φ 2 2 10 ( ¯ ˆ 1 F F √ √ ˆ s d d − (e The reduction of a subset of type-IIB supergravity on d (e D where It is not hard toeq. see (3.7) that on reducing a the circleother seven-dimensional produces hand, theory a the whose subset same lagrangianby of subset is Romans’ reducing of theory the type in Romans’ IIA six theory dimensions. on in On six the dimensions was obtained field. The lagrangian anddualizing equations of the motion 3-form of field the are six-dimensional theory [9] after any solution of thecan six-dimensional be and uplifted the to either five-dimensional type-IIA gauged or supergravities type-IIB supergravities. sense as T-duality.will However, show that for type a IIA on particular subset of type-IIA and type-IIB, we theory on a circleThe gives ansatz the of five-dimensional reduction theory from type-IIA without supergravity U(1) on gauged coupling. 4. Duals of 3-dimensional In this sectiontheory we on study a thestood torus. supergravity as dual The follows. ofwere gravitational flat, a system the Let 3-dimensional we isometries us of are this consider dealing system with would be can SO(1 be under- the NS-fivebrane. ItIIA is on not clear that a supergravity obtained from reducing type JHEP06(2001)025 . . . 3 R R S .On (4.1) 3 (which (in the S SU(2) SU(2) 3 3 S × × S L D ’s correspond to 1 manifold is given . In the table be- SU(2) SU(2) R 3 reduction, whereas 2 S 2 G × → T R 2) , , where the lower indices . In this case, the twisting 9 , 3 8 9 . In the UV limit the global , , S 3 7 8 R , 3 7 S SO(4). There is also an addi- S × tranverse to the fivebrane direc- 789 . The non-trivial part of the spin × 2 geometry. The 6 3 , SU(2) 2 T 5 1 , i.e. the directions 5 and 6. S 6 , T 6 × 2 , × R 5 2 5 3 T D × 56 T SO(4) × S ) 3 1 4 × , so that the four scalars transform as the 1 4 8 S → R 1 4 R R SU(2) × R × 1 × , so that it gives a twisted . The 3 , R 2 3 D , ) representation of SO(1 , 2) 3 SO(4) 1 SU(2) 2 1 , , 3 S 1 , × × S 1 × , 123 L 5) 2 1 0 , R 1 0 0 R SU(2) 8 of the are preserved, which is related to SO(1 / × ) and there are also 16 supercharges. After the twisting we get 2 → , 5) , 2 , 9) , 1 SU(2), one of them being the spin bundle of × Now, we focus on the twisting preserving two supercharges. As already men- the other hand, the normal bundle to the NS-fivebrane in the From the table 4.1, one can see that the NS-fivebrane is wrapped on the the time and the radialthe coordinate, geometry respectively. of In the ten form dimensions, ( the solution has belongs to the five-dimensionalother SU(2) two gauged spatial supergravity directions metric are ansatz), wrapped while on its representation ( 2 fermions (which aremetry) the transforming in two the supercharges ( of the remaining unbroken supersym- There are no scalarstwisting. after twisting, Therefore, while 1 we get one vector field as it is before the Therefore, the supersymmetry willthe be NS-fivebranes realized have through two directions atwisting. on twisting. The a brane Also torus, worldvolume notice so is that that on these are not involved in a It leads to a diagonal group SU(2) the up-lifting to 10-dimensional theory is obtained through an tions. Since the NS-fivebranes are not flat but wrapped on a second symmetry is SO(1 connection on this worlvolume is the SU(2) connection on the spin bundle of label the coordinates.supergravity Recall is from related the to previous the section five-dimensional that one the through a seven-dimensional resulting is SO(1 five-dimensional Romans’ theory),isometries we SO(1 have the following chain of breaking of the tional isometry group correspondingdirections to of the the fivebrane torus, are wrapped. wherethat we On the consider the two other here additional hand, hasthe the a spatial supergravity five-dimensional global solution metric SO(4) symmetry, has and the its corresponding ansatz for low, we schematically show thefirst global five structure coordinates of are thefor arbitrarily the ten-dimensional chosen Romans’ metric. to theory. The represent the five-dimensional metric tioned above, there are three spatial directions of the NS-fivebranes wrapped on sponds to the Lorentzone group is on the the corresponding flat rotation fivebrane group worldvolumes, of while the the second by SU(2) consists in the indentification offactors the in SU(2) the group R-symmetry of group the of spin the bundle fivebrane, with i.e. one SO(4) of the JHEP06(2001)025 2 R G × (4.4) (4.6) (4.5) (4.2) (4.3) 3 S × R and the torus get 3 S , k , σ are only dependent of the 2 3 ∧ Ω φ , d j i 2 σ r σ ) (4.7) , 0 ijk −  φ 2 dr . =0 1] part of the fivebrane will reduce to − )+1] dr k ∧ 3 ) r − ) σ ( r S 2 r ( dt ) ( w ∧ 1 φ r ) ( j , such that they satisfy the Maurer-Cartan ( r 9 M 3 σ ( =[ and the 2 3 w [ S µ Q − Q ijk r 2 1 2  , dx on dt + w i i µ )= + ) i )= r , σ i r A ( r ( σ ( ν f M dσ 2 = ν ∧ .Noticethatherewehaveadoptedthemostlyminus , e . We consider the non-abelian gauge potential compo- i 3 i ν S A σ = dw i 2 5 σ = ’s, together with the radial coordinate are embedded in a ds i 3 = F S 2 3 Ω . From the equation of motion of the dilaton the relation d r is the metric on 2 3 Ω d In order to show explicitly how the theory flows to a 3-dimensional SYM theory and it can be written as so that they are invariantSU(2) under gauge the combined transformations. actionand of The the is corresponding SO(4) given field rotations by and strength the is purely magnetic All the rest of functions, i.e. while for the abelian gauge potential we consider a purely electric ansatz where and consequently signature. In order tointroduce define the the left-invariant forms field configurations on this geometry it is useful to a point. This is in contrast with the fact that the transverse nents written in terms of the left-invariant forms fixed radii. Itin shows the that radial when coordinate onesmall on moves and the to no gravity the loweffectively dual) energy IR far three massless of in of the modes the the IR gauge are the dimensions excited theory gauge become (flowing on theory very this ison three-dimensional. two-space. a torus, Therefore, we brieflyRomans’ describe supergravity the presented SO(4)-symmetric in solutionstatic [13]. field of configuration, the Following invariant that under five-dimensional the reference,rotations. SO(4) let global As symmetry us we group already consider of mentioned, a spatial the metric ansatz has the structure radial coordinate manifold. In this way, sincebe our zero, IR it limit implies corresponds (as to we set will the radial see) coordinate that to the thefactthatthetwo equation JHEP06(2001)025 (4.9) (4.11) (4.12) (4.10) (4.13) .These H , , 2

ζ )+ 1 r , 18 ( +2 w 2

6 w w 1) + − + − ] (4.8) 2 2 3 − H ) ζ 2 r ) ( w 3 . 0, we obtain the expansions . ( )+ w w Vwζ r , 3 2 , ( 4 → , 4 ··· ··· w r , − − − + 6 2 ··· + ··· 2 )=2 ··· 3 2 H r 1) − r + r ··· + 1) ( + 3 2 ) − r V 3 2 +( 2 r + − r 2 r 4 ( 2 2 13 7 r ζ w r 5 w 1 w ( 288 13824 24 144 ( and 10 2 1) [2 7 7 2 ζ + + 288 − 288 ζ 3 − /r r r 1 r ] 2 + ) ) ν +2 r 1 r w 0 ( 96 )= ( 2 ( ν +12 φ r )= )=1 5 )=1+ (  4 e r r V r M ν ( ( ζ ( w )= 2 ζ to be 4. Also, we get 2 )= r p w M r − ( − V ( is an integration constant. On the other hand, from the  )=exp[ H Q  φ V r 0 )= ( 2 φ M r ζ ζ M ζ ( 2 1 3 3 1 ζ is an integration constant. The rest of equations of motion Q 6  − H = = )= ) r r ) r r r ( ( ( log log M dζ dw d d Since we are interested in the IR limit, i.e. when wherewehavedefined while for the dilaton we obtain In this case we have taken of the functions defining the metric,fields the for magnetic non-abelian the and five-dimensional the ansatz. electric abelian They are and straightforwardly can be obtained, where equation of motion of the abelian field, the following result with the field configurationreference and [13]. the metric The ansatzthe solution described supersymmetry must above, transformations satisfy can for the besuch gauginos equations found that and in obtained the gravitinos by following eqs.(2.3) setting first and order to (2.4), differential zero equations are obtained is derived, where equations are compatible withfive the dimensional lagrangian equations given ofdifferential in motion equations section preserves derived 2, two from and supersymmetries. the any solution Romans’ of these first order JHEP06(2001)025 3 , 0. 2 , 3 1 S → (4.15) (4.17) (4.14) (4.16) 3 , 2 , 1 )+ 2 , R 2 dZ dY remain finite as + φ 9 2 6 , 8 √ 15 , 8 3 7 dY . S ( , φ +e  6 6 , ) 2 2 3 5 √ 3 + 5 , T r , and in the IR limit ( dZ 2   , 2 3 φ ) O +e T ) 6 3 Ω 4 9 , √ r  + r d 8 3 ( remain finite, while , 2 3 ( 2 2 2 − , 9 O r Ω r 3 , 1 O 8 d , S + − 7 +e 2 + 21 2 r in the five-dimensional metric ansatz 2 2 576 2 R r  r 3 i 1 − dr to zero. It is obvious from eqs. (4.15) 6 − S 2 ) 7 involving the coordinates 1, 2, and 3, the √ 0 , 1 r 5 960 and 11 gA 2 φ ( / 3 dr 1  ) 0 T S ) i 1 − φ − 0 M r φ R i 1 13 ( Firstly, we consider the case when the solution Now, we consider the metric given in eq. (3.14) 1 6 σ 3 gA −  √ 0, M 0 2 − =e − φ e 2 dt i 3 3 − 3 =1 2 , → i √ ) 2 X σ √ 2 g , 1 5 r ( 2 1 ( r 4 φ ν dt 6 R 2 3 =1 ) 9 i √ e r =e = X 8 (  ν 9 φ − 2 T , 2 6 e 8 φ e , 3 R √ φ. 6 2 7 2  7 √ − g 1 6 R 8 e 4 φ e 3 √ 6 2 4 √ = 0. Futhermore, the above solution can be up-lifted, following the + 13 − g φ. 1 5 4 6 r e = = √ + − ˆ φ 2 10 ˆ s = = d ˆ φ 2 10 around ˆ s d r atze presented in the section 3, to either type-IIA or type-IIB theories. In these we can see as follows radius is given by Since the type-IIB NS-fivebrane is wrapped on and (4.16) that in the limit so that we can see how the radius of the shrinks to zero in the IR. On the other hand, the radii of effectively reduces to adefined point, on the in torus. this limit weUp-lifting obtain to a type-IIA twisted theory. gauge field theory ans¨ two cases, the IR limit turns outUp-lifting to be to the type-IIB same. theory. In this way, one cancore, see while that both in fieldorder the strengths, IR i.e. limit the the abelian non-abelian and gauge the potential non-abelian has one a are of the is up-lifted to type-IIB supergravity. From eq. (3.12) the 10-dimensional metric is Therefore, using the previouslythe calculated different IR manifolds. expansion Thus, we for can the obtain the radii of Without loss of generality, we can set JHEP06(2001)025 are are 1 9 ’s (in Q (4.22) (4.18) (4.19) (4.21) (4.20) 1 S ), while S ,andin g 1 9 S (2 and / 8 , 7 , and 3 6 and also =1 1 S 5 8 , S , V 7 , 1 5 , 6 , . , 3 S , R w 2  ,  3 2 3 1 ) . S 3 Ω , r and d  ( 9 ) 1 ζ  3 O ) R r , 3 − ( r + ) = ( 2 4 ζ. O 2 r 5 r O ( dζ p + R 5 O , + A solution with no electric abelian 2 2 ζ ζ 1 r 2 2 27 + ζ 8 r 24 4608 ζ / 2 24 1 / r − √ 27 − e 1 0 2 4608 e 1 φ 0 105 12 4608 6 0 r  7 − r √ 8 0 ζdt = 1 to zero. It implies that φ =  6 1+ 3  9 √ r ) √ =e  H 8 0 2 ζ 3 φ φ/ − 0 , 6 2 e φ 2 , 9 √ (12 6 √ 2 1 2 8 / √ e 15 1 g 1 8 R − e 4 2 0 r =e = =e = 0, in the IR, the radii 0 2 2 8 5 9 = , 7 φ -symmetric solution. R , R 2 5 2 6 ds R effectively shrinks to a point as in the type-IIB case, we get the 3 , SO(4) 2 , 3 1 S is an integration constant. The metric is given by = 0, so that the radii of the torus (in type-IIB case) and the two 0. Since the type-IIA NS-fivebrane is wrapped on 0 0 r φ → 3 In addition, in both cases one can use the criterion for confinement given in ref- , 2 , 1 fields can be obtained by setting zeros, as we expecteddifferential since equations no (4.9) electric can field be are easily integrated, excited. yielding In the this relation way, the first order the IR limit R finite Using the criterion of referenceis [4] not it acceptable, is both straightforward to in see type-IIA that and the type-IIB IR theories. singularity which shrinks to zero in the IR limit. In addition, the radii of same geometric reduction aswhen in the previous case. Note that this can be obtained type-IIA case) are exactly the same. erences [26, 27], in order to show that the correspondingThe static potential singular is confining. Therefore, as we did for the type-IIB case, we can obtain the radius Again, by considering In adition, for the dilaton we have the following relation where JHEP06(2001)025 after 2 S even when it is = 1 super Yang- 00 g N = 1 super Yang-Mills. = 1 super Yang-Mills N N 13 = 2 super Yang-Mills theory. Actually, this solution is N = 2 SYM in three dimensions. In these cases, the fact that N = 1 super Yang-Mills theory on a torus. N We remark that the singular solution obtained in [13] is produced when the considered in the ten-dimensionalrepresent any theory. gauge field This theoryknow means if in the that the presence IR. this of We electricleading solution think abelian to does that fields is a it not related would desingularizationdeserves to be a of further interesting rotation the of to investigation, the solution. weThe fivebranes, issue can Although, of discuss the we resolution here thinkone of that a of the singularity this little the can point about cases bethe understood studied this three-dimensional as in mechanism. follows. [9],related We recall to which the has one been given in interpreted [5], as and the it represents gravity a dual smeared of NS-fivebrane on T-duality. What iscase worth was stressing produced is by thatof the the eq. excitation resolution (4.2), of as of non-abelian weto the fields. have turn seen, singularity In on in in the the order this case electric to abelian of obtain the fields. a metric non-singular We metric also it have is to necessary recall that for that particular From the point ofof view the of Romans’ the five-dimensional supergravityIIA theory theories, can supergravities. the be SO(4)-symmetric up-lifteddimensional solution These to supergravities, either are respectively. type-IIB obtainedconsists or It through of type- NS means the fivebranes, that either up-liftingpaper the type [9], to IIB 10-dimensional we or seven constructed system IIA. atained and On in non-abelian the references six- solution other [28, that 29]. hand, isIn in It this identical our has case, to been previous it interpreted the was in solution a [5] ob- gravity as dual a of wrapped a NS-fivebrane. theory very similar to electric abelian fields are turn off. This solution has a singular 5. Discussion In addition, in [9] we interpretedtheory the massless as solution the of the same Romans’the NS-fivebrane six-dimensional with IR a theory compactified was direction. In such a situation, Mills theory on a torus,Although which many is aspects confining. Thisconsidered of IR [30, three-dimensional theory 31, super is 32], interestingtheory Yang-Mills some on on theories its aspects a own. have of torus been can three-dimensional are be still an interesting poorly motivation fordual understood. further of studies Therefore, since the we have results presented a obtained gravity here their actions inpergravity the and string the frame correspondingmass, are seven-dimensional indicates similar, one that with for vanishing thosefive-dimensional the topological are supergravity massless the studied six-dimensional same here,involves su- system. NS-fivebranes. since We again Thus, can from thethat see the 10-dimensional this a analysis system is similar in thedimensions. issue the natural in present In extension the paper, fact, of we in the conclude the six IR and limit seven-dimensional this results case to corresponds five to JHEP06(2001)025 , , 04 ]. we 2 r (1998) 253 super yang 2 =1 ]. on Antorchas of N hep-th/0007018 J. High Energy Phys. , ]. ]. Supergravity duals of gauge ]. ]. ]. limit of pure (2001) 822 [ hep-th/9802109 N Gauge theory correlators from non- Adv. Theor. Math. Phys. A16 , . We think that this fact induces the Fivebranes wrapped on associative three- 2 M-fivebranes wrapped on supersymmetric r (1998) 105 [ hep-th/0011190 hep-th/0012195 hep-th/9711200 14 hep-th/0008001 hep-th/0011288 to be 3 B 428 S Towards the large- . It would probably render a similar situation as r limit of superconformal field theories and supergravity Supergravity description of field theories on curved man- ]. Int. J. Mod. Phys. (1998) 231 [ nez, , nez for collaborating in the early stages of this project (2001) 588 [ N 2 (2001) 008 [ u˜ (2001) 106003 [ (2001) 126001 [ nez, 86 Supergravity and D-branes wrapping special lagrangian cycles 03 Phys. Lett. , gauged supergravity in six dimensions D63 D63 ]. (4) The large- F hep-th/0103080 Anti-de Sitter space and holography Phys. Rev. Phys. Rev. nez, I.Y. Park, M. Schvellinger and T.A. Tran, Phys. Rev. Lett. , , , hep-th/9802150 Adv. Theor. Math. Phys. critical [ ifoldsandanogotheorem mills cycles J. High Energy Phys. cycles theories from (2001) 025 [ [1] J. Maldacena, [2] S.S. Gubser, I.R. Klebanov and A.M. Polyakov, [3] E. Witten, [4] J. Maldacena and C. Nu˜ [5] J.M. Maldacena and C. Nu˜ [6] B.S. Acharya, J.P. Gauntlett and N. Kim, [7] H. Nieder and Y. Oz, [8] J.P. Gauntlett, N. Kim and D. Waldram, [9] C. Nu˜ write a more generalin function [5, of 9], i.e. the resolution of the singularity with only non-abelian fields. 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