JHEP05(1999)003 May 6, 1999 Accepted: 3-branes. D February 14, 1999, = 1 superconformal theories dictating their 3-brane and discuss the near-horizon super- of coincident D N N Received: ∗ D-branes, Supersymmetry and Duality, Duality in Gauge Field This paper lays groundwork for the detailed study of the non-trivial superconformal theories = 2 superconformal field theories by mass terms for adjoint chiral fields. [email protected] [email protected] [email protected] N =1 HYPER VERSION On leave of absence from St. Petersburg Steklov Mathematical Institute ∗ CERN, Theory Division, 1211and Geneva Department 23, of Switzerland Physics,New Yale University Haven, CT 06520E-mail: USA Institute of Theoretical and Experimental117259, Physics Moscow, Russia, and Lyman Laboratory of Physics,Cambridge, Harvard MA University 02138, USA E-mail: Lyman Laboratory of Physics,Cambridge, Harvard University MA 02138, USA E-mail: Abstract: renormalization group flow connectingsions supersymmetric using fixed string theory pointsat on in AdS singularities four spaces. of dimen- Specifically,and Calabi-Yau we have threefolds consider an D3-branes ADE which classification. placed low-energy generalize dynamics The are the infrared fixed conifold pointsing singularity arising ADE from deforming the correspond- We probe the geometry with a single Samson Shatashvili Nikita Nekrasov Steven S. Gubser Generalized conifolds and fourN dimensional gravity solution for a large number Keywords: Theories, Brane Dynamics in Gauge Theories. JHEP05(1999)003 , 5 11 13 X × of con- 5 B with four- 5 AdS ). In gauge S N × SO(6). The low- 5 ≈ large. AdS N 2 YM g = 2, 1, or 0 superconformal N Γ Y 1 Γ Y and the ’t Hooft coupling -branes at the singularity: Gauge theory 15 D N of N D3-branes placed at an orbifold singularity of a Calabi-Yau N -brane on ALE space-brane at the generalized conifold singularity 7 3 Γ, with Γ being a discrete subgroup of SO(6). The corresponding = 4 super-Yang-Mills theory with gauge group SU( D D / 5 S N = 5 X The conjecture has been extended [4] to the spaces of the form 2.2 Single 2.3 Relation to2.4 the geometric invariant Construction quotient of the Kahler metric on 9 2.1 Single 2.5 Properties2.6 of the metric on Geometry of the base of the cone3.1 Large number 3.2 The supergravity3.3 geometry and chiral Blowup primaries Modes and RG flow 16 14 19 provides fascinating possibilities for the study of both sides of the equivalence. dimensional where symmetry according as Γ is aenergy subgroup dynamics of SU(2), of SU(3), or SU(4) three-fold is described by one of these gauge theories. gauge theories have been described in [5]. They have 1. Introduction The recently proposed duality [1, 2, 3] between string theory on a space stant negative curvature andB certain gauge theories whichThe live original conjecture on [1] the identifies boundary type of IIB string theory on Contents 1. Introduction2. Gauge theory perspective 3 1 3. Supergravity and holography4. Dual constructions with branes5. Conclusions and conjectures 15 20 22 theory terms, the validity ofory the depends supergravity on approximation having to both type IIB string the- JHEP05(1999)003 2 Z =1 =1 / 5 (1.1) S N N A, D, E U(1) was conifolds. is an arbi- / =2 5 M N ADE SU(2)) × where 5 M conifold: × (SU(2) 1 5 A ≈ . ; then figure out the field the- 5 1 , AdS 1 M =0 T 2 × T 5 = 5 + case there is a moduli space of such mass 2 1 M AdS 2 Z A type, and we will call them + 2 Y ADE + 2 X over the complex plane; our singular geometries will be fibra- 1 A was [6]. There the space 5 S = 2 orbifold theories. Unlike the = 1; our field theories descend from mass deformations of the general N 3-branes which are placed at the conifold singularity are described by an The purpose of this paper is to construct holographic dual pairs out of an infinite The first successful example of this approach for a manifold not locally diffeo- In general one could consider string theory on N superconformal field theory whichmalization is group. a non-trivial While infrared thisworked fixed work out was point in in [7] of progress, using the toric a renor- geometry. further class ofclass examples of was conical singularities.ALE The space geometry of (1.1) type is a fibration of a four-dimensional type ND ory of D3-branes movingknown close singularity, to work that out singularity.three-branes from In on practice, supergravity the we the singularity, may construct near-horizonthe start a geometry singularity, with gauge of and a theory black consider describinggauge the D3-branes theory near result is asnificantly worked a distort out holographic the in dual geometry. thecoupling, pair. This approximation whereas approximation As that supergravity is a the issuperconformal, correct rule, valid we D3-branes in the may at do the feel strong limit not confidentthe coupling. in of comparison sig- extrapolating weak with it If supergravity to canpergravity the strong down be gauge to coupling made weak theory so coupling directly. that is is string The much theory harder “extrapolation” because in of it a su- requires background the with full Ramond-Ramond type IIB fields. morphic to tions of ALE spaces of arbitrary considered, which is the base of what we will call the deformations which is isomorphicversal to deformation the of projectivization the of corresponding the singularity. moduli In space all of cases the we will have superconformal symmetry, which is one quarter of maximal (eight real supercharges). trary Einstein manifold. This Freund-Rubinnear-horizon ansatz is geometry the with most only generalmension the static bosonic metric five is and the the firstwhich self-dual are where five-form not there excited. even locally are Di- diffeomorphic, infinitelycorresponding and many field a different natural theories question Einstein to are. manifolds suggests ask is a The what two all D-brane the step originisolated approach of singularity to the such finding that holographicthree-brane the conjecture the located near-horizon answer: on geometry first this in singularity find supergravity is of a a manifold black with an orbifold theory with mass termsto for chiral fields added to break the supersymmetry The field theory constructed in [6] descends by RG flow from the JHEP05(1999)003 k E and k the Kahler D k A cases we fall back on k E Γ, where Γ is a discrete / 2 and C case) of our gauge theories by k k D conifolds. However, we exploit a A ADE 3 , we present an explicit construction of the Higgs k , we show explicitly how the conifold arises from the D k A or 3-brane. k D A on the conifold geometry to show that the spectrum of chiral ∗ C -brane on ALE space -brane placed at the orbifold singularity D Γ, in the case where the orbifold is not deformed. In the case where it is D conifold geometries are non-compact, but they can all be realized as sin- / 2 C ADE In the section 5 we present our conclusions and some conjectures. In section 4 we present the realization (in the In section 3 we outline the supergravity side of the dual pair. Writing out Section 2 is devoted to the study of D3-branes near the orbifold singularities from First of all we recall the construction of the gauge theory on the world-volume conifolds, but on many points we include also the analysis for the k deformed, we show how thecomplex deformation parameters two-form. are For related Γ to = the periods of the using other branes as a background instead of the singular geometry. the formal argument presented earlier. Finally, we calculate for Γ = primary operators in themass-dimension relation. gauge We theory then is proceedmodes with correctly and more the reproduced detailed analysis corresponding by of AdS the the supergravity blowup holographic multiplets. metric from gauge theory atthat the it classical level, is exhibit not its theagreement cone Calabi-Yau with structure metric. and the We observe results then of briefly [8]. discuss the reasonsexplicit for that, metrics in for theare not Einstein known spaces in seems closednatural form impossible action for since of the Calabi-Yau general metrics branch, 2.1 Single 2. Gauge theory perspective In this section we describethe the CY geometry threefold of obtained ALE spaces byconstruction and fibering is present ALE obtained a space construction by over of world-volume looking a at one-dimensional of the base. Higgs a This branch single of the gauge theory on the which our conifold theoriesgeometry descend. are claimed Iftheory, to then the the be D3-branes first thing described moving thatto should by in be a the verified a is infrared single given that thisquestion. singular limit D3-brane gauge theory We of has specialized present a a for formalsingularities. particular its argument why gauge moduli For this should Γ space be = precisely so for the the ADE singular conifold geometry in gularities of compact Calabi-Yau three-folds.A Our results arecases. most complete for the Our solution of the F- and D-flatness conditions; for the of a single JHEP05(1999)003 1 1 2 (2.3) (2.4) (2.1) (2.5) .Letus i . . . µ of each ver- 2 i n 22 ), while a lower i n 2 3 1234321 1 1 7 ). We will suppress k i E D n is the complex moment . This number coincides = 2 supersymmetry. Its i i n µ i N . 12345642 P i 11 β 8 j = E γ 1 2 321 The extended Dynkin diagrams of . . . ji h 1 , for the extended Dynkin diagram, , 2 B ) i i j . ij φ γ 1 n s i i α 6 µ .Let U( i =0 ij E 111 r Tr i B =0 k 4 r i µ ij i A tex. Figure 1: ADE type, including the indices s X × Tr = 0 corresponds to the trivial representation. j = = i X i X 1 ’th are adjacent nodes and the sign, which is part of = W i G (2.2) i j β ’th vector multiplet and i i α i and µ i )ofthe . j j ¯ n r indicates a fundamental representation of U( , ij bi-fundamental i i : a i n α µ ij C 6=0givesriseto of Γ. The label j is determined from a 1when i ⊕ ij ± ij r is a complex matrix, is simply the dimension of a a i = = = 1 point of view each ij n i r B ij N s is the scalar of the indicates an anti-fundamental representation of U( ⊗ with runs through the set of vertices of extended Dynkin diagram of the corre- ). i 2 j φ ji i α C ), where ’th gauge groups. The theory has a superpotential: j i, j j The matter content of our gauge ¯ ,B n , ij i n B sponding ADE type (seerepresentations figure 1), or, equivalently, through the set of irreducible where Here an upper index with the dual Coxeter number of the corresponding ADE Lie group. where such that index the decomposition: hypermultplets in the representations ( transforming in the ( a pair of chiral( multiplets, call them From the theory is that of subgroup of SU(2) ofgauge ADE group type. is the The product: field theory has introduce an antisymmetric adjacency matrix map. There is some gauge freedom in the choice of explicit expressions for The number these color indicesrelation when among their the contractions are clear from context. There is one our gauge choice, indicates a direction on the edge between them. Then we can write and pair JHEP05(1999)003 (2.7) (2.8) (2.6) ’s) ALE i ,whichis ζ 1 G for various Γ as has periods which 0 ζ S ’s coincides with the S i ζ 1) : of h ζ − +1 ω k k x, y, z 2 z 2 . 2 z 2 . z + 2 2 i 12( z z + . 2 h/ φ + ) 2 action, which is specified by giving 2 + + +1 − 3 y xy Tr , C ∗ 3 3 y i k k as follows: ]= ( i ζ y y i + + z n C n + z 1 i + + Id y +1 − X i k k 4 3 5 Γ into the (smooth for generic ][ 5 GL ζ / i x x x x x y − 2 and 2 = × 2 = = = = = +1 with coordinates C − i =[ W = 3 µ k k Γ Γ Γ Γ Γ ). Since all the matter is in the bi-fundamentals, x, y i β f f f f f c 1 C → n G ] : : : : : W x 6 7 8 k k 1 2 34 6 12 46 9 18 6101530 E E E A =[ D 2. = 2 supersymmetry one may introduce complex FI terms α h/ N k 6 7 8 k Γ (it follows from the complexification of Duistermaat-Heckmann E E E A D ζ =1+ = 0 are invariant under a =0inthespace to the coordinates β Γ Γ f f + α α, β . It can be described as a quotient of the space of solutions of the equations ζ = 0 the Higgs branch becomes the singular orbifold, and at the singularity S ζ It is important to notice that the holomorphic 2-form We would like to list here the equations which describe Without breaking The equations Notice that embedded diagonally into each U( hypersurfaces The moduli space of vacua (Higgs branch) of the theory with which modify superpotential as follows: For the Coulomb branch appears.massless At vector the generic multiplet, point which of corresponds the to Higgs the branch U(1) there subgroup is of one by the complexification of the gauge group it is neutral with respect to this U(1) subgroup. space depend linearly on theorem [9, 10]). This observation will be used below. the weights complex deformation of the orbifold JHEP05(1999)003 . x π + for y )= 0 j x (2.9) t ( x (2.11) (2.10) 1 ’s gen- + ,...,k i − x x k 2 and y =0 : i ,R : ) x ,i x x x, y C ( . integrates to 1 ) − z . the rational curve k )=0 . The two-form is 2 i is determined by ) i x x 2 x r -plane, whose fiber is xR ( i>j z 4 dy/ x ( , 6= +1 π + i>j − k ( 2 x 2 0 2 P t , . We now wish to relate the 2 2 2 xy , . z z z k +1 α< m l )= + , ζ D − x for such an y j k )+ )+ )+ ≤ m 1 0 . ’s as the Weyl group of the type , = x ζ x x x t i − l x ) as polynomials in i X ( ( ( j m ζ t t is the rational curve 2 1 4 5 4 -plane which connects ,0 − dz is described by the equation = ; ’s. The fiber over its generic point z )=0.For − π x P P i i k x X =0 Q α )+ m +1 . To get a non-trivial period of it we 0 √ =1 ∧ x x k l + x 2 = S + + π Γ ( x, y + z 2 dx cos π/ ( P 2  dy y y y are the roots of y df r j − f ) ) ∧ ) 6 i k is ∧ x x x )= x )+ z x = ( ( j ( )= Q t ’s act on 2 dy z )+ 2 3 √ i 3 x ; x z 2 dx x x ( Q P Q + ( − , = k − 1 i = i α + + + x, y ζ +1 Q − dy/ x ( , hence its periods are: x ζ k where k 3 3 3 ω , ( z f 1 i ω P x y y y sin √ 2 r π −  x / ` be its roots: ∆( = = = = = ) − . = 6= = iπ i k ∗ Γ Γ Γ Γ Γ dy x x x f f f f f ij y ` = ] C t ∧ , ζ .) Hence we get 2 ij 1 ω ] for : : : : : r [ − =1 ζ dα dx ` ∗ k 6 7 8 k k 1 2 ’s. In order to do so we study the periods of the holomorphic ω l = [ ). Let C E E E P A ζ D factorizes as: x =( 2 = = 0. Consider the discriminant ∆( ( + z ζ ζ 2 2 z k ω ω − z 2 + x and k 2 k Q )+ y and doesn’t pass through other t In this case the ALE space is a fibration over the In this case the fiber over the point dy/ x )= + ( j x 1 1 is isomorphic to ( . are some coordinates on the base of deformation. There are canonical for- − k − 6= k k x k t P (write i given by: erate the action of the Weyl group of the type C contains a non-trivial one-cycle, over which the form The branching of the square roots in (2.11) and the permutations of x The permutations of the roots R The form A case. case. isomorphic to choose a one-dimensional contour in the The period of the one-form hence The complex deformation of the surface k k A D mulae, listed, say in [11], which represent where coordinates two-form, where JHEP05(1999)003 ∗ = E C )+ 2 over z x ( j (2.13) (2.12) 3 .Soin x )+ j being a A ]shrinks x x j ( φ ). By con- x [ B and µ ( and i + − Γ )=4 i y x ] i ) x Y x x x ( requires inverting l A ζ + . 3 with respect to the y and action described in (2.8) h =0 k ∗ t 2 C z . series. 2 + . z under the action of the element ! and therefore vanish in the limit t cases we can be explicit: dz + ; 2 h 2 h A, D k t β ∧ ! y µ t dx , φ D planewithfiberovergiven ) ; , of the discriminant ∆( dy x β these one-dimensional contours span a Γ φ α ( y i x φ with the conifold-like singularity of the ∧ φ γ x φ , and is the elliptic curve dF j 7 Γ

over which the zero-cycle [ α i x k x x dx Y x Γ φ ± ij Z A f ∧

φ h -plane with the generic fibers being isomorphic Γ φ = 1 theories whose Higgs branch coincides with x f dφ h ≡ N φ is homogeneous of degree ) Γ = Ω= is the one-cycle which vanishes both at F µ . It is endowed with a holomorphic three-form: x )isfiberedover ) cases or elliptic curves (with infinity deleted) in the C µ ( , φ,µ Γ ( z φ, x, y, z ζ 2 dy Y ( S Γ x is fibered over let us fix a one-dimensional contour connecting A, D F C ) H φ,µ has weight 1. ( φ, µ ζ )= φ S x -brane at the generalized conifold singularity in the ( In these cases the fiber over ). The periods are given by D γ ∗ x ( C 2 B . Now let us turn to concrete examples of this case the identification between the coordinates 27 the elliptic functions. 0 and it degenerates over the roots cases. where ∗ 0. Indeed, the function plane which connects two points The space First of all we need to show that these manifolds have shrunken three-cycles. Let C k φ → E ∈ two-dimensional surface. The nontrivial three-cyclein is obtained if we get an interval to zero (i.e. the points collide). In the which the one-cycles vanish. As we vary following type: iff the variable 2.2 Single We are going to show that its periods scale as Let us call the non-compact manifold described by this equation as cases. For given struction the manifold action in (2.8). Therefore the form Ω scales as µ t us deform the equation (2.12) to the non-compact Calabi-Yau manifold We now proceed with describing to either The equation (2.12) is homogeneous with respect to the particular ALE space JHEP05(1999)003 (2.15) (2.14) (2.16) 0. . → =0 µ ! . x 2 0 φ t

ξ , . 1 4 , 2 i , ) +1 t φ 0 2 3 k ; − . w ξ dw is nothing but the Higgs branch of w dw Tr ) ( i i t ξ ξ n . Γ ; +1 m i σ q ξ Y k Id n I ( 1 2 1 φ P = 0. By varying with respect to the mat- ,P I i Id − 2 i +1 1 k φ k 8 = X m +1 1 − µ k k µ R dW − = − must generate a trivial gauge transformation φ 2 µ +1 i k = i − = − φ = φ 1 W w i are: − . Consider the curve: µ ! k we get: . Thus the role of the mass vector is to choose the → ± ij x φ i w φ i φ =0. φ y/φ

W i m = +1 m k = i P i and they also vanish in the limit ,η ζ 2 P z, 2 x/φ / 6= 0 plane with the fiber being the (generically) smooth ALE ) = 0 (it follows from (2.5)). The space of solutions to (2.16) is = . Then the period of the three-form Ω reduces to φ dy i ξ ∧ m is a non-trivial one-cycle on the curve 0 all these periods clearly go to zero. i x/φ n σ Let The conditions on dx i = → ξ P µ =( ζ = 2 theory described above perturbed by the superpotential term: where The periods of the three-form Ω reduce to: Let As case. case. ω Indeed, let us look at the equations Now we wish to show that the manifold N k k A D ter fields we get the condition that space, corresponding to Then, varying with respect to the which is only possible when: direction in the moduli space of ALE spaces of given ADE type. with the only condition The necessary andprecisely sufficient condition for the equations (2.16) to be solvable is fibered over the JHEP05(1999)003 t ) ), Y V [11]. V,V (2.19) (2.17) (2.18) group, f )bethe ∗ g C ( γ Hom( )beanyiso- ∈ α x, y ( Γlet X f ∈ -valued functions on g onto this orbit. This ) by the action of the C 2 Y . Consider the matrix 2 , , ,X ) 1 C − 2 are the exponents of J X i , 2 ,X d z 1 1 cases explicitly) that the Higgs − ) = X ( g 2 (called Milnor bundle). Its fiber D ( f )is: γ 2 Y where α = -thespaceof has a natural action of the } X 1 and ,X i Γ ) h T 1 d − action. Restrict g ) { C ,X A ( 9 X g ∗ È in the two-dimensional representation of Γ. γ ( + = C γ J )=0. g ) W t 1 = Γ. 2 ; z ]=0 V ). Consider the space of pairs β 2 ∈ examples in some detail. we have: = are constructed as follows. Let ,X X g x, y 1 k β 1 f Γ ( ,X Γ. To this end we slightly reformulate the solution α V,V 1 be the base of its versal deformation. The dimension X Y f / X g ( )ofthe ,D 2 X k f T [ φ action described in (2.8). The space of orbits of this ) C ( A t g which obey two conditions: ∗ ( ) for all C γ = V g ( t γ ) be any Γ-invariant function on 2 ,z = 2 theory in the case where all deformation parameters are zero is the surface 1 z . It depends on the choice of orbit, that is on the choice of mass ( T Γ . N i f comes with the canonical bundle Y ∈ In this case the solution for ( Γ. m t are the matrix elements of T -flatness conditions. Let ∈ ).Duetoinvarianceof β 2 F g α 2 of operators in is called the Milnor index of the singularity. It also coincides with the rank of g case. , ,X T 1 Now let us look at the In other words, the spaces Now let Choose any orbit k X =1 A of ( α The space f Our space is the quotient of the space of these pairs ( for the 2.3 Relation to the geometric invariantIn quotient this section we wish to show (in the which originates in the where group of gauge transformationswhich commute (cf. with [12]). The latter are the elements of End( parameters the group Γ. Thisinduced is for naturally concreteness a by representation the (called left regular action representation) of of Γ, Γ on itself. For corresponding element of Hom( over point for any lated simple singularity. Let r the corresponding ADE group. The space action is a weighted projective space is nothing but the orbifold branch of the is our space JHEP05(1999)003 , k , 2 2 . z (2.21) (2.22) (2.23) (2.20) (2.24) 2 1 z ,which Id 2 ,...,ω 2 z = 1 Z· z Z . = ,ω,ω Id )= , , 2 Z  , ! . Z· 2) ,X − +1 1 − Φ k 2 J = k 0 1 z X 2( 2 2 ( M t + iz z = X J k Z 1 ) − ]= Y J = . 0 − . 2 , 2 ( 1  z − 2 − +1 , − ,X k

k 1 1 Φ z Id = Φ, then the superpotential is 2) Z ,X = X . k − 1 M [ ) = k 2 Id − 1 2 Y· ,J +1 ) 2( 1 − k X g z 2). The basic invariants here are: −        ( Y· 4(  Z . Taking the trace of the last equation ] have a block diagonal form: 0 0 0 γ , )= 2 2 − φ 2 − = 2 z = 1 k )Φ 2 ,X ,X z ...... 10 g ,X +1 ,X 1 2( ( ! 1 k . . .1 1 2 which we sketched at the end of section 2.1. ]=0 = γ XY X 2 ZX + × X . . . X Γ ( J Y 0 Φ[ Y 2 . . . = ,X 010 001 100 2) Y  2 iz singularity:        [Φ , − Y ,X + k 2) k = J 0 − A 1 =Tr , Id k + z ) is diagonal matrix with entries being 1 2( 2 , J Id g is 2(

z ( W X· k γ ) ± = X· J = − 1 ]=0 = 0. The space of solutions to this equation modded out by the 1 provide the most efficient way of making completely explicit the X -type equation: +1 +( k )= k 2 1 . The basic invariants are: M 2 ,X D 2) X for some complex number ,X +1) − [Φ ,X | 1 k k 1 ( Γ | 2( 1 X X z πi/ In this case the matrices ( Id 2 φ e = X = X where the size of which obey in the basis where obey the equation with ω case. where The corresponding matrix functions are clearly scalars: It is obvious that k D abstract construction of the spaces The matrices The requirement of F-flatness is The first two expressions intion: (2.24) are Φ satisfied = when Φ is a trivial gauge transforma- If we introduce a third matrix Φ with in (2.24) tells us Tr complexified gauge group (which implements D-flatnessshould along be with the gauge invariance) generalized conifold. JHEP05(1999)003 . . ,i is Γ ,i = 1 Y − 1 i +1 B i ,the b b X (2.28) o (2.27) (2.26) (2.25) { k , +1 + for some , i,i J .Bycon- b i φ }        Ξ m . Γ =diag . . . +1 0 0 ) − = i,i 1 φ , where all fields b ,ω,...,ω i i Γ X ξ } { ξ 1 ... x ...... 2 n Id 6=0caseitmaybe 2 . − − . . . x X 0 x ,i which take us all the 1 x m ( 1 − = 1 . . . i 0 00 +1 ,...,r x i + 1 products ξ ± 2 +1 =diag =1 and b ¯ Y i k k . As a subgroup of SU(2), i,i        d X 1 b } } ,i =0 1 )=diag i , with (2.27) being the only i ,i 1 X = i φ X +1 )= . g m Q i | } +1 ( 2 { ) i where β α Γ γ b ,i db X = )is ! } 1 conifold equation from and 1 i 1 +1 +1 + ± Γ ξ i − 1 X k − c 0 x b i,i { Y g b ; ω ( A +1 { ,...,x =diag γ i,i 2 0 +1 ω 0. We assume that the metric on α b ¯ i,i d

M ,x X . Defining the gauge invariant quantities ≡ 1 ) i ,b 1 +1 i )=det( x = 11 2 g m φ { i,i ( β =diag ( X +1 Φwemusthave α γ { db =1 ) 2 i j r 1 M , so indeed the moduli space has three complex = = + X g P ( 1 ± i β B c )(det ¯ =diag φ − case. X 1 ]= X d , we recover the i 2 k β X 2 = =1 Γ i α A i ) dφ } X ,X i ξ 1 ,and 1 g x k =0 φ and a matrix Ξ = diag = 0 the metric is exact, while in the i X { X , =(det x x = m =det − 2 to be an excess of gauge-invariant products parameterizing the c .Wehave − c } + ds =diag c ,i , 1 } +1 i i X b x { { apriori Let us use the notation . To satisfy [ } 1 has as its generating element =det diag diag − k − + i,i vention we may take complex number way around the extended Dynkin diagram, there are A dimensions. It is easy to understand thisseems point how the F-flatness conditions eliminate what In a basis for the regular representation where for diagonal matrices. So for instance J c But these may all be expressed in terms of b case. general solution to ( moduli space. In addition to the products equation among The original space of fields is k A 2.4 Construction of the Kahler metric on Of course, in the case affected by the quantum correctionspotential. which lead At to the anyquotient renormalization rate construction of we the in shall Kahler the describe the metric which one gets by the Kahler Gauge theory gives us an explicit construction of the Kahler metric on the space are complex and weflat: have identified JHEP05(1999)003 . 3 Ê (2.32) (2.33) (2.31) (2.29) (2.30) (2.34) , 2 ’s in terms of the ) b i i ω φ i . + m i . . =0 = dθ ( )=0 )=0 2 iθ ,i i ) i | 2 1 i r ,i =0 e φ +1 − ¯ x 1 φ i i

t i , + b − . φ i ξ − ,i i 1 2 i b +1 =: − r =: i − +1 ,x +1 i i,i i +1 i i d~ k X i i i i,i t k b b t , i −| α b φ t

x φ φ 1 φ, i r i 2 ( ( 1 r | obeys: ξ − − ,i = = 1 =( =2 = = = i i +1 ,i − i ?d , − +1 2 ,i ω α ,i i,i i | r ~ i i,i 12 ··· ··· b b +1 +1 +1 x ,i b i = +1 +1 + | i m b i,i i i,i i i b = = +1 b b = b ¯ i − θ + +1 b d 1 1 i is taken with respect to the flat metric on +1 dω t 2 φ x i,i | = ? +1 i,i b 7→ ,i −| b = 1 i = i,i 2 − θ 0 +1 | i 0 i t db ξ -flatness conditions, which means that we restrict the b φ +1 F + − i,i −| i b ,i | ξ 2 | +1 i b ¯ +1 -and d i,i ,i D b | +1 i ), where i db ,θ , and the Dirac connection i | i r ~ r ~ | are the complex numbers which are uniquely specified by the following -flatness conditions imply that: = is three-dimensional and i i ξ d r D, F metric (2.28) onto the surface of equations: where conditions: The gauge group acts as follows: The where We now impose the where It is convenient, following [13], to rewrite the flat metric on coordinates ( We have: The projection along the orbitsnals of to the gauge the group orbit. is Formally achieved this by taking is the equivalent orthogo- to the following procedure [13]: replace JHEP05(1999)003 1 ± are i,i b ± (2.35) (2.37) (2.39) (2.40) (2.38) (2.36) (2.41) c case it is k .Theresult A i , , A 2 ) ) A φ . l . ξ 2 + )) | +1 φ iθ − φ l k l dθ ξ ξ x ( − ( e 1 − d | − ). − etc., the variables ¯ φ φ x x x ∧ i i   V | ξ ξ ) 2 | +1 φ + + l − − φ t, x, is written out as follows: i,i i 2 ξ ¯ (arg( φ b ξ t x x , Γ +1 | iθ d k − 2 2 Y q − 2 1 2 + | | e |   . x i  φ φ x 1 2 1 ( 2 i i ξ 1 dc | on | | d  ξ ξ W dφd ± 2 − φ ∧ | + i i,i $ k − − + + ξ 2 =0 φ b 2 l i X | i c t 1 x x dφ ξ ξ φ | | i − k 4 l 13 =0 q Y i ∧ ξ and minimize with respect to ¯ − x φdx + + | + Γ 2 = 2 2 − x ¯ dx | t t Y ¯ φ + Ud 1+ t ds ± x d | 2 q q c +   t − ∧ 2 + ¯ t k k k x q =0 =0 =0 Ω= i i i X X X ?dV , 2 t dφ q + i 2 = = = = q t + is taken with respect to ( t − Udφd   U V d   W dA )+ l − X A = = 2 r 1 2 , compute the ,i + +1 . In terms of the coordinates -flatness conditions a choice of the gauge for the phases of +1 +1 = Vd~ i,i i i dθ iz b b ( A A = ± ∧ 2 D, F − comes equipped with the holomorphic three-form. In the y i dt ds Γ A Y = = + ± $ c dθ by i given by the formula: expressed as follows: where is the following metric: where With this choice of phases the Kahler form has to be made. We choose: The space dθ where In solving the 2.5 Properties of the metric on where in the last formula JHEP05(1999)003 can ¯ φ (2.43) (2.46) (2.47) (2.42) (2.45) d ∧ .Now,the dφ A . It is invariant $ dt ∧ . which is a part of the ¯ φ. dθ ! d . + 2 = 1 orbifolds. =0theterm . This action is generated 2 ∧ | ∧ | Ê | α V ¯ U φ φ N | φ i d | dφ . dφ 2 ξ +1 = 2 case, here the geometry as k − ∧ ∧ ∧ = − ¯ Ω (2.44) N | + ¯ x x W dφ dx | x d ∧ θ |

∧ ∧ ∧ + Ω = ¯ 7→ x 2 κ  log t t d dx ¯ ∂ = ,θ q ∧ ∂ ∧ Vdt iα 14 i $ 2 1 = X dt dx xe ∧ j ¯ 2 1  ∧ 2 | )+ $ dz 7→ + . Moreover, the Higgs branch becomes a cone over U A ∧ 2 ∧ + | ,x i + iV dθ φ −| $ Ê | iα dz − 2 1 j ¯ dθ i φe ( i = R VW  ¯ Ω=  7→ 3-brane and by the fundamental string is different — unless ≡ H 8 3 ∧ φ D as the symplectic manifold with the two-form R Ω = Γ . Explicit computation shows that : 3 Y κ $ which is in turn a U(1) bundle over four dimensional Kahler manifold 5 M For the Kahler metric to be Ricci-flat it is necessary that: The fact that the metric doesn’t come out in the Ricci-flat form may sound trou- Hence we are led to believe that, in contrast to Another problem is that different terms in the formula for the metric (2.35) Let us think of which we now describe in some detail. -symmetry (2.8). holomorphic three-form turns out to have rather simple form: for some constant under the U(1) action a fivefold the assumption (2.28) that thethorough initial discussion, Kahler and metric also was [8] flat for is more wrong. examples See of [14] for a bling. On the other handThe it reason seems is that that itweak we does in study not the abelian receive infrared gauge quantum so loop theory all corrections. and loops the must coupling go inobserved away. by this the theory single is and the Kahler form (2.40) scale differently under the by the Hamiltonian: B be neglected. As ainvariant result under of “RG” action this of “RG flow” the metric on the Higgs branch becomes 2.6 Geometry of the baseNevertheless, of close the to cone the singularity where R Of course, another choice of gauge leads to the gauge transformed From this expression we get the volume form: and (2.44) is not obeyed. Instead, for the Ricci tensor we have: JHEP05(1999)003 } µ = ) i T (3.2) (3.1) R : is the (2.48) (2.49) subject 5 . Z by 1 : 3 i M È defined by r Ê Y × Γ : ⊂ Y 1 2 X , defined as the È ( 5 + 1). In the case Ê { k = 0. For example, M ∈ ( = Γ x 3 − ,~ F 3 bythesameamount È W Ê ζ ∈ → y and . , W | y, ~ 2 i φ | φ x, ~ ~ . , ) (3.3) =1 | i ) Tr i 3-branes placed at the orbifold point x 2 i | | Γ D φ m Nn ↓ t, i Y B, Nn 1 2 = 0 that is the quadric ξ 2 i U( SU( − → i i T X 15 x × × | + − 2 defined by the equation = + = U(1) Z W 2 coincident . To be more precise, consider the following “RG 2 h N t N ζ ˜ + G G q N → = 1. It is easy to show that this space is isomorphic 2 can be described as a complex hypersurface in the = ,α,β, | i 3 1 Y y ~ X W 0. As a result we get the manifold -branes at the singularity: Gauge theory hypermultiplets in the bi-fundamental representations: B H È + + D ij W 2 → x ~ a | X of µ + 0 by the action of the U(1). The base of the cone is described as a quotient of the subvariety in , with the metric (2.35) where | is the set of pairs of vectors N y ~ B B − ζ> x ~ | = t, θ, x, φ H = 2 theory describes in agreement with the expectations about the conifold geometry (cf. [6]). ). case we would get the hypersurface in the ordinary 2 j N 1 . The gauge group is now: S i Γ. We now perturb this theory by adding the mass term: A r Nn × / , 2 i 2 In the infrared limit the U(1) factors decouple and one is left with the gauge In general the manifold This ⊗ S C = 1 the manifold N Nn to to the condition the equation hypersurface in the flow”: perform the simultaneous rescaling of level set of the Hamiltonian: k The matter fields are the where the manifold It is clear how toC proceed with generalizations: replace the vector spaces 3. Supergravity and holography 3.1 Large number It is easy tofiber show using bundle: the metric (2.35) that this action is also free. So we have a group defined by the equation weighted projective space ( and then take the limit in in the space of JHEP05(1999)003 i 2 N m ’th ,z C 1 (3.5) (3.4) N z anomalous makes ij B M  ) . N 2 i 2andthe by the dummy space φ ,z / i Γ N 1 µ on which the allowed gauge z C ( , 2 Tr β (1) ,z α =0 1 i dx z X m , which is what we expect. α : 4. This is the result we actually will i Γ / Y dx 1 + ,...,X αβ 2 i − ) g 1 2 µ 2 r ,z 16 Tr 1 1 i are taken to zero (but not all, since the + z i ( 1 m 2 2 m (1) α dr 6=0 X i =  X m 2 6 anomalous dimensions of 1 : i i . ds φ = Γ Y =diag -tuple of the parameters eff ) are non-zero, it is straightforward to see from the condition N N i W ( α m X 4. Typically the number of independent constraints is less than / have anomalous dimension 1 ij − B now have the following form: 3-branes placed at 2 , 1 We now proceed with showing that UV superpotential yields the moduli space Methods described in [15] can be used to determine the possible anomalous di- Indeed, the discussion of the section related to the geometric invariant theory are invariant over this space. Thus we can calculate these dimensions at the point X ND ij transformations act as the group Γ does. Turning on the mass matrix use in comparisons with supergravityto predictions. continue to By continuity hold we as would some expect of it the dimensions of the number of independentsolutions. anomalous When dimensions, all so there is actually a space of on the superpotential that theB dimensions of all gauge-invariant combinations of the where all the symmetric product of the generalized conifold live in the deformed ALE space. It means that the Higgs branch looks like the so they depend on an which is a trivialfor representation of the group Γ. As a consequence, the expressions are only defined upwould to be an overall surprised rescaling). toinvariant We find operators. do a not continuous have spectrum a of proof possible of this, dimensionsof but for we gauge and the effective superpotential: 3.2 The supergravity geometry andLet chiral primaries us start byat briefly an reminding isolated singularity the ofa reader a Calabi-Yau the six three-fold dimensional supergravity and manifold that versionof [16]. the of D3-branes Assume metric D3-branes can the near be manifold the written is singularity as before the addition mensions at the infrared fixed point.of chiral Linear fields constraints result on from the setting anomalousing the dimensions that NSVZ the exact superpotential beta be functionsin to dimension zero our 3. and models These demand- by constraints can giving always the be satisfied goes through with the only change that we now tensor JHEP05(1999)003 , = x , x =0 (3.6) (3.8) (3.7) ± c f ) E + , 3-branes are 2 C , +1)and∆ ds relates to chiral : k 2 r =0 ( ND / 1 3 4 f r∂ !  4 = 4 r L ± 2 c 1 r part of this action is the 1+ . When + + :∆ 5 1

r -charges listed in (2.8). Since Ê ± ∂ : exactly the eigenvalue under M + 5 R i,i E r b  r 2 3 . ∂ 5 5 dx 4+ 1 r -symmetry group. From the existence M  √ + R κN 2 2 = Vol 2+ dx f π 17 ) 2timesthe − 2 d / + to denote the five-dimensional Laplacian on √ ∗ . The U(1) part of the action is an isometry of 2 1 d = λr dx + 4 ∗ to the cone! To be more precise, only a subset of f + L → f 2 is the gravitational coupling. Supergravity is a good dd f r 4 dt 0 conifolds, we can consider the complex variables − α =∆ =( k = 0, the supergravity metric is 2 s  g f f A 2 r 7 ) r / ) ¯ 1 is much bigger than the Planck length and the string length. ∂ correspond to chiral primaries: these are the operators which π ∗ r∂ − ¯ L ∂ ! 4 4 + which are also eigenfunctions of the operator r L =(2 ∗ charge with dimension held fixed, and the eigenfunctions they are f ¯ ∂ ¯ R ∂ πG 1+ action on the conifold geometries. The 2(

∗ is an Einstein metric on a five-manifold = C =16 = ∆(∆ + 4). By considering a complete set of harmonic functions on the 2. These dimensions are just 3 2 αβ 2 / g E κ ds In particular, for the This pictures applies to the conifolds (2.12) as follows. In (2.8) we specified an A complete set of harmonic functions on the cone can be generated from har- The holographic correspondence as worked out in [2, 3] relates on-shell fields in of the harmonic extension =3 , and in the field theory it is realized as the r 5 Φ where ∆ dual to are in fact the ones which extend to holomorphic functions on the cone. the eigenfunctions of wherewehavereservedthesymbol maximize U(1) Here 2 action of approximation when and Φ as holomorphicfrom functions. their Near representations the as IR products fixed of point the we fields know their dimensions of the Calabi-Yau metric on thefive-dimensional conifolds Einstein we are manifolds. learning Unlike ofthese the the existence Einstein coset of manifolds manifolds a class constructed havein of in moduli following [17], [18] spaces. isFortunately To impractical there discuss because is chiral we a cannot more primary efficient write operators way, down which the we metricmonic will functions explicitly. now explain. dilation symmetry of the cone, M where placed at the singularity primary operators with dimension ∆ = the base of the cone. Together the two conditions in (3.8) imply that ( r∂ where cone one can extract the full spectrum ofthe the bulk five-dimensional of scalar spacetime Laplacian. to operatorsthe on arguments the used boundary. in In [18], the present the context, spectrum following of the scalar Laplacian JHEP05(1999)003 1 5 A are M (3.9) r r∂ cases. k E , and Φ agree x and , k ± c D K, The cube of the Kahler form 1 =2 K  fields separately in a product of the i which defines the conifold. We have z j i z ∂ B log ¯ -charge. The arguments of the previous action on the conifold geometry, of which ∂ R ∗ 18 . i + and C z , and Φ, one can verify that the gauge theory i i z . In the gauge theory they reside in long multi- x which have a definite eigenvalue under 5 , A ∂ i case by showing explicitly that F-flatness con- ± log case) supergravity predicts in addition on order of M z c 1 ∂ 5  A S i ,andΦby ∆ x . The constraints are more complicated in the general l i , j X which are homogeneous with respect to the weight in (2.8), ± B c i l chiral primaries with dimension less than ∆. As in the i z ¯ Ω, where Ω is the complex three-form. So Ω has dimension 3. 3 ∧ ; hence so does the Kahler form. ...A r 2 j B 2 i A 1 j B 1 -charges are determined by the conifolds the analysis is equally straightforward for the i case, but the conclusions are the same: because the moduli space (namely R k A is one generator, we have shown that the dimensions of It is worth emphasizing that holomorphic functions on the conifold were intro- Holomorphic functions of the A We note in passing that an appropriately chosen Kahler potential should also have dimension non-chiral fields of dimension less than ∆. These come from the non-holomorphic 1 r 5 is proportional to Ω Finally, writing out Ω in terms of just polynomials in the argued in section 2 thatmodulo the solution complexified of gauge the invariance, F-flatnessification results conditions of in in the the this gauge gauge same theory, the invariance conifold. gauge implemented theory The D-flatness. complex- aresuperfields, Now, constructed modulo chiral from primaries F- sumsmension. in and of It D-flatness gauge follows conditions, invariant thatwith products and chiral the of with primary homogeneous chiral operators definite holomorphicchecked are functions conformal in in on di- one-to-one (34) the correspondence conifold. of [6] This was for essentially the ∆ r∂ identified modulo the equation relating the between gauge theory anddirect supergravity, way up to to fix that anunder normalization overall dilations is normalization. of to note The that most the metric (3.5) has dimension 2 dimensions indeed agree completely withthe supergravity. Although we have focused on form ADE the conifold) is parametrizedmatter fields by modulo holomorphic the gaugecombinations F-flatness invariant represent conditions, combinations are the of precisely chiral primary the the fields holomorphic which functions these on theduced conifold. as a trick to write down efficiently the eigenfunctions of the Laplacian on two paragraphs show with minimalright calculation dimensions that and holography degeneracies predictsThere for exactly chiral the are primary on operators order in ∆ the gauge theory. case [18] (but in contrast to the which minimized dimension for specified these straints allowed one to symmetrize 2. Written in terms of the complex variables, this amounts to the homogeneity condition eigenfunctions of the Laplacian on where we collectively denote JHEP05(1999)003 : ` 2) | 2 , (3.10) =1tensor ` = 2 means sixteen in (3.10) can be 2 N X independent 2-cycles. 6= 0 an antisymmetric k = 2 theory, deformed by ` N 4 2 3 3 2). | 2 , . The Kaluza-Klein reduction = 1 fixed point with a quartic 1 S N introduces × X 2 1 2 5 = 4 gauged supergravity in five dimensions. S µν λλ F = 0; and for X F N on one of these cycles gives rise to an AdS Γ were discussed in [19] (see also the ap- dA / 19 . Both these equations of motion are valid ∗ 5 in string theory on a manifold which is (at least . The set of scalar mass terms corresponding S d 2 j U(1) operator dimension + MNPQ i`A of − 0 4 2 2 MN × A − 1 1 3 1 1 TrΦ B S = define − 2 i , blowing up the SU(2) dA satisfying k ∗ E µ = 4 supergravity in five dimensions to find such solutions . We do not have a complete enough understanding of the A field N times a compact manifold through a gauge field theory which 11 scalar scalar scalar tensor ,or k 5 T D , satisfying k AdS and leads a tower of fields labelled by the Kaluza-Klein momentum U(1), and the types of gauge theory operators they are dual to, are A 2 1 µν × Z S A However, we can at least describe a multiplet which plays a key role. / 5 2 on S =0avectorfield − MN Blowup modes of the fixed For a globally supersymmetric conformal field theory in four dimensions, ` B 2 real supercharges, which is theThe same supergroup organizing number as the multiplets in in both cases is SU(2 of pendix of [20] forto a indicate more precise how discussion thisflow. of analysis For Kaluza-Klein feeds Γ reduction). = into Our the aim supergravity is interpretation of the RG explicitly. Lagrangian of gauged tensor field at anti-self-dual two-form potential The gauge theory operators we have identified schematically as written more precisely as TrΦ The self-dual four-form potential asymptotically) plets whose dimensions are notis algebraically no protected, good and aspredictions. understanding far for as we why know the there dimensions should3.3 match Blowup the Modes supergravity and RGIf flow it is indeed true that one can group theory. The multipletmultiplet. we will The be bosonic particularly components,group interested their SU(2) in quantum is numbers under the the R-symmetry as follows: mass terms as insuperpotential. (2.14), to The a simplest non-trivialbetween case infrared would be a supergravity geometry interpolating lives on the boundary,supergravity the then full we renormalization group would flow expect from an to see reflected in some solution of only at the linearized level.are termed Tensor fields “anti-self-dual” which in satisfy [21],anti-self-dual the where antisymmetric latter (among tensors equation other and of things) of motion the vectors superpartners are of worked out using SU(2 JHEP05(1999)003 ) 1 1 5- S S =1 × corre- 8 kM , N 1 1 =0and − i Ê ` θ =2super- 3-branes are − i of the highest = 1, form the N θ ` ` ND Atheoryon modes in (3.10) at the Á 5-branes become 2 -theory, where the circle X =2with M B side. The corresponding Á kNS N . Their world-volume is 01236. and 3-branes placed at the orbifold 1 . The differences -duality along the 6’th direction. S k λλ T ND = 1 that enters into (3.10). and the ` operators in (3.10) are the correspond- ,...,θ 1 10 in the 789-plane. The 1 S θ r ~ λλ 20 × 1 6 S -fieldontheType =4with = supergravity [21]. It contains a pair of scalars of 2 5 type then we get the Type N T 1 − k AdS A NSNS B =2 N 4-branes which wrap the circle 1 and a spinor (left- or right- handed) of U(1) charge zero. The U(1) ND ± -duality between the ALE (more precisely multi-Taub-NUT) space and T 5-branes, whose world-volume is 012345 (6 being the coordinate along multiplet of 5-branes are located at the points kNS NS fluxes become visible if the whole picture is lifted to The relevant part of the multiplet (3.10) which is turned on forms what is called Part of the analysis of [19] was to determine the dimensions of these operators, is promoted to the two-torus spinor 1 6 anti-self-dual tensor multiplet of 2, together with an anti-self-dual tensor multiplet of spin state. Thus in particular, two right-handed spinor multiplets with If the ALE singularity is of linearized level. One wouldextract need a a concise solution description of ofUnlike the the the full relevant RG interactions nonlinear flows to theory,lating considered perhaps between in along the those the orbifold papers,throughout lines and the the of the supergravity flow. [22], conifold solution [23]. shouldeight At interpo- preserve supercharges, the four respectively. UV real and supercharges IR endpoints one shoulda recover sixteen and The U(1) charges charge assignments can be shifted by the Kaluza-Klein momentum spond to the fluxes of the and which are locatedmapped at to the the same point RR S singularity of the ALE6789 space. are the Let coordinates 0123 of ALEto be space. the the Let U(1) world-volume 6 isometry of be of the the compact the 3-branes, direction ALE corresponding while space. Perform conformal theories in this language. Consider with 4. Dual constructions with branes Most of the constructions whichcan we were be studying redone using the into geometry the use or the language field theory of branes, along the lines of [24], [25]. The idea is supersymmetry. listed in (3.10), from theThe masses first of step the corresponding inand modes finding conifold in a geometries the should supergravity tensor be solution multiplet. to interpolating turn between on the the orbifold fivebranes. First let us remind the reader of the realization of the ing fermion mass terms which are turned on at the same time to preserve to all the independentperturbations 2-cycles introduced of in the (2.14). blown The up orbifold form a basis for the mass JHEP05(1999)003 . =1 (4.1) πj/k N 4-branes =2 j θ ND 4-branes are no D 5-branes one finds .The 2 ’th brane its position 5-branes to be frozen NS i T . NS 2 ) = 0) and leads to φ T i ξ of a D3-brane charge. This is rather than taking be another holomorphic coordi- − describing the relative positions π j 2 x ζ θ j ( / C r ~ i on the torus ) = 1 Q k NS − 8 − . i B i i θ r ix ~ + φ i F the corresponding coordinate. Then the 0 + − ξ 5 i 7 ,...,z 21 4-branes. The tilting makes the scalars in πα θ 1 x = 2 ix z D e i direction causes the type then in addition to x + Z 4-branes has the meaning of the Coulomb branch φ 4 5 D D x : iµ 5-branes in the 45 directions. Another way of seeing φ = 5-branes in such a way that for φ NS = 2 supersymmetry massive, since the Aside. NS Á N 5-) branes are dislocated in the 789 plane then the correspond- in the 789 plane and vice versa. Of course, this is the familiar 4-branes in the M of the NS B-field. Consider wrapping a D5-brane around one of r 2-cycles which sum to zero in homology and through which there ~ D 1 5-branes which wrap the whole of the k − i θ 5- (or − NM orbifold singularity which has been resolved by FI terms: the geometry i . θ NS 1 ~ ζ − k A in the Wess-Zumino part of the D5-brane action [26], Finally, let us consider what happens when we add D5-branes to the stack of Now let us rotate some branes. Let If the ALE singularity was of The plane 45 transverse to the If the D3-branes on an orbifold singularity. For simplicity we will restrict our attention NS this is to noticevector that multiplet the is condition identical that to FI the term equation is (2.16). proportionalN to the scalar into the an are lifted to direction. Let us denote by at the same point picture of the transitionsFI between terms. the Coulomb and Higgs branch with or without This configuration of fivebranesthe preserves fivebrane supersymmetry “wrapping” (one a can holomorphic think curve of it as of a special case of thein phenomenon [12, of 27, fractional 28], branes only and here wrapped we branes are discussed allowing arbitrary ing ALE space is resolved. The parameters these cycles, with itsB other dimensions in the directions 0123. The term linear in orientifold plane on Type gauge theory on the world-volume of longer free to slide along the the vector multiplets of nate. Consider tilting the in 78 plane linearly depends on of the fivebranes inspace. the The 789 corresponding plane processFI are in terms the mapped field to theory the is hyperkahler described moduli by turning of on ALE the separation of the gives this wrapped D5-brane precisely ( branes which are located at the points is a direct productplane of the (dimensions 4-dimensional 45), ALE andabove, space there flat (dimensions are Minkowski 6789), space the complex (dimensions 0123). As remarked are fluxes JHEP05(1999)003 is a UV , Γ Y considered UV , Γ [5]. The six- -fields through Γ / Γintoasmooth Y B 5 / 2 = 2 gauge theory S 3-branes are placed C and there is a flux of D N 5-branes wrapped on a 5 UV ’th gauge group becomes . The manifold D i M C × × 5 . The properties of the string Γ number of N / 2 AdS N C to ); now the = as in the absence of any orbifold. = 2, and the hypermultiplets are still N 5 UV , UV S , Γ N Γ Y 22 × Y 5 × 3 , 1 3-brane placed at the singularity of the threefold. AdS Ê which is equal to D + 1 complex marginal deformations corresponding r factors of SU( Γ. When the large 5 UV This has the effect of changing the gauge group: it was / = 2 gauge theory of the ADE type which corresponds and the fluxes of the RR and NSNS k 5 3 M case in [25]. Unfortunately, at the moment it does not τ S N 1 = A ), with 5 UV N M SU( ’th NS5-branes. = 2 superconformal theory has a number of interesting deformations. i N ×···× + 1). The supersymmetry is still ) The It would be nice to T-dualize back from brane realizations of gauge theories to N N We thank A. Karch for bringing to our attention the reference [29], which includes a similar 1’st and 3 − cone over the base the collapsed two-cycles which are fibered over the fixed circle in as a Higgs branch of the gauge theory onIt branes. is known thatto it has the exactly couplingsspace-time of dilaton+axion various gauge factors. Their space-time counterparts are the RR five-form field through deformed by the generic FIinterpretation terms of flows this to fact is the that trivial if IR one fixed first point. resolves The the orbifold space-time dimensional tensor multiplet which contains theseof fluxes the also deformations contains the of parameters thewould two-cycles themselves correspond (three to parameters per the cycle). FI These terms in the gauge theory. The space and then places thethe large near-horizon number geometry of threebranes will at be the generic point of it then seem to be very practical. We start with theto quiver the manifold which locally looks like 5. Conclusions and conjectures So far we described afold. class Our of complex construction threefolds ison which most the generalize world-volume easily the of described ordinary the in probe coni- the language of the gauge theory Upon T-dualizing, the D5-branei becomes an extra D4-brane stretched between the SU( discussion of this construction. at the singularity they canspace-time no geometry longer from be that treated of as probes. Instead, they change the theory propagating in thissuperconformal background gauge theory are which occurs believed at to the origin be of the reflected space in those of the SU( 2-cycle as modifying awas gauge suggested in theory [30] by basedT-duality incrementing in on the a evidence rank perturbative from of D-brane anomalous setting one brane reinforces creation. gauge that group The interpretation. obtain use the of exact supergravity/stringto background the which are construction dual for to them, similarly in bifundamental representations. The interpretation of JHEP05(1999)003 1 in A =2 =2 5 IR N M N × 5 Γfixedbythe / AdS 5 S supergravity which 5 = 2 origin of the gauge AdS N some fields in this multiplet 5 ). We have taken one more step AdS 11 is the base of the cone described T 5 IR in the ultraviolet and M = 1 chiral multiplets within the . 5 N UV 23 5 IR M M × 5 = 2 supersymmetry) to its IR fixed point (with and N AdS 5 UV M Γasabove,and / metric should be recovered at either end of the flow (al- 5 5 S has the same topology as = 2 AdS Z / 5 5 UV S M sufficiently large and include the matter multiplets arising from the twisted N Assuming such a string background exists, what are its properties? It should have To put it in a single phrase, the two ends of the RG group flow correspond to There are two distinct claims one could make regarding D3-branes located at =1). theory begs the questionorbifold in singularity and what “flow” sense to one the can conifold start geometry. with D3-branes It at was an argued ADE in the action of Γ.ADE In orbifold a singularity, nutshell onemass can the deformations turn second and on claim in fieldsstring is the which that theory string in theory starting background the are which with gaugefrom twisted tracks D3-branes theory its sector the are at UV modes, RG the an fixed andN flow point obtain which a (with takes the gauge theory the rotational andand translational it symmetries should of preservevalid four-dimensional description four Minkowski of real space, the supercharges.take low-energy dynamics Five-dimensional of supergravity bothsector is ends of a of the the flowflow. orbifold), (provided It so we is it not(as clear seems was whether likely done truncating in thatfixed the effect it theory points. in to is [23] a The in small and number fact [22]) of is valid multiplets a throughout controlled the approximation far from the the infrared. includes the blowup modes and observing that in case in [6] that therearesolutionof is no topological obstruction to the flow (more specifically that have just the right tachyonicinvolved masses in to correspond deforming to theanalysis the gauge scalar of and theory. [19] fermion from masses The the states twisted sector in localized this at multiplet the arose circle in on the though with differentfull radii, ten-dimensional related string to theoryapproach the description the we central expect factorized charges to form as see in the metric [18]), smoothly and in the in section 2.6.complex structure The we totalwhat described space on this in of coneRG the the should flow. previous cone play the sections. is role the of Calabi-Yau It a manifold, radialthe is two coordinate, whose not Einstein dual manifolds, clear to scale to in us the toward describing such a flow by identifying the multiplet of the Calabi-Yau singularities we havegiven described. a conifold The singularity first ofbranes and a simplest located particular is ADE at as type, follows: that the singularity low-energy theory is of the D3- IR fixed point arising from an theory deformed by giving masses to the vector multiplets, as described in section 2 and 3.1. 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