AdS_4/CFT_3 duals from M2-branes at hypersurface singularities and their deformations
aa r X i v : . [ h e p - t h ] S e p October 31, 2018
AdS / CFT duals from M2-branes at hypersurfacesingularities and their deformations Dario Martelli and James Sparks Swansea University,Singleton Park, Swansea, SA2 8PP, U.K. Mathematical Institute, University of Oxford,24-29 St Giles’, Oxford OX1 3LB, U.K.
Abstract
We construct three-dimensional N = 2 Chern-Simons-quiver theories which areholographically dual to the M-theory Freund-Rubin solutions AdS × V , / Z k (with or without torsion G -flux), where V , is a homogeneous Sasaki-Einsteinseven-manifold. The global symmetry group of these theories is generically SU (2) × U (1) × U (1) R , and they are hence non-toric. The field theories maybe thought of as the n = 2 member of a family of models, labelled by a positiveinteger n , arising on multiple M2-branes at certain hypersurface singularities.We describe how these models can be engineered via generalized Hanany-Wittenbrane constructions. The AdS × V , / Z k solutions may be deformed to a warpedgeometry R , × T ∗ S / Z k , with self-dual G -flux through the four-sphere. We showthat this solution is dual to a supersymmetric mass deformation, which preciselymodifies the classical moduli space of the field theory to the deformed geometry. ontents d = 3, N = 2 Chern-Simons-quiver theories . . . . . . . . . 42.2 Vacuum moduli spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 IR fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Parent d = 4, N = 1 theories and Laufer’s resolution . . . . . . . . . . 10 k = 0 . . . . . . . . . . . . . . . . . . . . . . . . 264.2 Adding RR-flux/D5-branes: k = 0 . . . . . . . . . . . . . . . . . . . . . 284.3 Brane creation effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.4 The field theory duality . . . . . . . . . . . . . . . . . . . . . . . . . . 36 T ∗ S . . . . . . . . . . . . . . . . . . . . . . . . 375.2 The deformed M2-brane solution . . . . . . . . . . . . . . . . . . . . . 395.3 The G -flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.4 The Z k quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 The Stenzel metric 53C A different reduction to Type IIA 54
The work of Bagger and Lambert [1] (see also [2]) has led to new insights into thelow-energy physics of M2-branes. In [1] an explicit three-dimensional N = 8 super-symmetric gauge theory was constructed, a theory which was later shown to be aChern-Simons-matter theory [3]. Following this work, Aharony, Bergman, Jafferis, andMaldacena (ABJM) [4] have constructed a class of three-dimensional Chern-Simons-quiver theories with generically N = 6 supersymmetry (enhanced to N = 8 for Chern-Simons levels k = 1 , × S / Z k , or their reduction to Type IIA string theory. This hasrenewed interest in the AdS /CFT correspondence, opening the way for the construc-tion of many new examples of this duality, in which Chern-Simons theories are believedto play a key role [5].An interesting generalization of the ABJM duality is to consider theories with lesssupersymmetry. For example, the case of N = 2 (4 real supercharges) is analogous tominimal N = 1 supersymmetry in four dimensions. In the latter case, when the gaugetheories are engineered by placing D3-branes at Calabi-Yau singularities the naturalcandidate holographic duals are given by Type IIB string theory on AdS × Y , where Y is a Sasaki-Einstein five-manifold. It can similarly be argued [6, 7, 8] that a largeclass of Chern-Simons-matter theories should be dual to N = 2 Freund-Rubin vacuaof M-theory. This duality, for toric theories, has been studied in many papers – see,for example, [9].In this paper we will discuss a three-dimensional Chern-Simons-quiver theory thatwe conjecture to be the holographic dual of M-theory on AdS × V , / Z k , with N unitsof quantized G -flux, where V , (also known as a Stiefel manifold) is a homogeneousSasaki-Einstein seven-manifold. This can be thought of as the near-horizon limit of N M2-branes placed at the Calabi-Yau four-fold singularity z + z + z + z + z = 0 , z i ∈ C , (1.1)which is clearly a generalization of the well-known conifold singulariy in six dimensions.Indeed, Klebanov and Witten mentioned this generalization in their seminal paper210], concluding with the sentence: “ We hope it will be possible to construct a three-dimensional field theory corresponding to M2-branes on (1.1) .” In the present paper wewill realize this hope. We propose that the three-dimensional field theory in question isan N = 2 Chern-Simons-quiver theory with gauge group U ( N ) k × U ( N ) − k , generalizingthe ABJM model. The matter content and superpotential will be presented shortly insection 2; see Figure 1 and equation (2.6).The supergravity solution possesses an SO (5) × U (1) R isometry, which reduces to SU (2) × U (1) × U (1) R when we perform a Z k quotient analogous to [4] with k >
1. Thisis therefore the first example of a non-toric
AdS /CFT duality. In fact there are veryfew examples of this kind, even in the more developed four-dimensional context. Thesingularity (1.1) is the n = 2 member of a family of A n − four-fold singularities, definedby the hypersurface equations X n = { z n + z + z + z + z = 0 , z i ∈ C } . Thus we arenaturally led to consider a family of Chern-Simons-quiver theories, labelled by n , whoseAbelian classical moduli spaces are precisely these singularities. Here the n = 1 modelis the ABJM theory of [4]. Naively, this suggests that each of these theories will havea large N gravity dual given by AdS × Y n , where Y n is a Sasaki-Einstein manifolddefined by Y n = X n ∩ S . However, the results of [12] prove that for n > do not exist . This means that the field theories we constructcannot flow to dual conformal fixed points in the IR. We will review the argumentfor this in the course of the paper. Nevertheless, we can study these theories in theUV, and in particular we can, and will, discuss their string theory duals in terms of aslight generalization of the Type IIB Hanany-Witten brane configurations [14]. Thiswill allow us to derive field theory dualities, in which the ranks of the gauge groupschange, using the Hanany-Witten brane creation effect. We emphasize again that theAdS Freund-Rubin solutions exist only in the case n = 1 (the ABJM theory) and n = 2.One of the motivations for studying these models is that on the gravity side thereexists a smooth supersymmetric solution which approaches asymptotically the AdS × V , / Z k background [15]. For k = 1 this solution is a warped product R , × T ∗ S , where T ∗ S denotes the cotangent bundle of S , and there is a self-dual G -flux through the S zero-section. In fact, the deformed solution corresponds to deforming the hypersurfacesingularity by setting the right hand side of equation (1.1) to a non-zero value. This A different proposal was given in [11]. However, this was not based on Chern-Simons theory. We note that it was suggested previously, incorrectly, that these singularities lead to AdS holo-graphic duals [13]. The solution is completely smooth only for k = 1. For k >
3s a complex Calabi-Yau deformation, precisely analogous to the familiar deformationof the conifold in six dimensions. Indeed, superficially this solution looks like the M-theory version of the Type IIB solution of Klebanov-Strassler [16]. In the IR the twosolutions are precisely analogous; however, in the UV they behave rather differently.In particular, the M-theory solution here is asymptotically AdS × V , / Z k , without thelogarithmic corrections which are a distinctive feature of the solutions of [16, 17, 18].The topology of the solution at infinity can support only torsion G -flux, but a carefulanalysis reveals that in fact in the deformed solution this torsion flux is zero. Thus weare led to conjecture that the theory in the UV is the superconformal Chern-Simons-quiver theory above, with equal ranks of the two gauge groups. We will argue thatthis solution corresponds to an RG flow triggered by adding a supersymmetric massterm to the Lagrangian. This was already observed in [19], but we will here describe inmore detail the deformation in terms of the superconformal Chern-Simons theory. Inparticular, we will see how the deformation of the field theory modifies the (classical)vacuum moduli space, precisely reproducing the deformation of the singularity (1.1).The plan of the paper is as follows. In section 2 we introduce the Chern-Simons-quiver field theories: we compute their classical vacuum moduli spaces and discuss therelation to parent four-dimensional theories. In section 3 we discuss M-theory and TypeIIA duals of these Chern-Simons theories. In section 4 we construct Hanany-Wittenbrane configurations in Type IIB string theory, and discuss a brane creation effect inthese models. In section 5 we describe the deformed supergravity solution. In section6 we identify this deformed solution in the UV with a specific supersymmetric massdeformation of the field theory. Section 7 briefly concludes. We relegate some technicaldetails, as well as a different Type IIA dual, to a number of appendices. We begin by describing a family of d = 3, N = 2 Yang-Mills-Chern-Simons quivertheories. The family is labelled by a positive integer n ∈ N , where the n = 1 theory isthat of ABJM [4]. d = 3 , N = 2 Chern-Simons-quiver theories A d = 3, N = 2 vector multiplet V consists of a gauge field A µ , a scalar field σ ,a two-component Dirac spinor χ , and another scalar field D , all transforming in the4djoint representation of the gauge group. This is simply the dimensional reductionof the usual d = 4, N = 1 vector multiplet. For the theories of interest, we takethe gauge group to be a product U ( N ) × U ( N ). We will therefore have two vectormultiplets V I , I = 1 ,
2, with corresponding Yang-Mills gauge couplings g I . To the usual N = 2 Yang-Mills action, we may also add a Chern-Simons interaction. This requiresspecifying the Chern-Simons levels k I , I = 1 ,
2, for the two gauge group factors. Theseare quantized: for U ( N I ) or SU ( N I ) gauge group k I ∈ Z is an integer. In this paperwe shall only consider the case that k = − k ≡ k ; for k + k = 0 the dual stringtheory description will be in terms of massive Type IIA [20], which we do not wish toconsider here.The matter fields of an N = 2 theory are described by chiral multiplets, a multipletconsisting of a complex scalar φ , a fermion ψ and an auxiliary scalar F , which maybe in an arbitrary representation of the gauge group. For the theories of interest,we consider chiral fields A i , i = 1 ,
2, transforming in the ¯N ⊗ N representationof U ( N ) × U ( N ), and bifundamentals B i , i = 1 ,
2, transforming in the conjugate N ⊗ ¯N representation. We also introduce chiral fields Φ I , I = 1 ,
2, in the adjointrepresentation of U ( N I ), respectively. This gauge and matter content is a quiver gaugetheory, where the quiver is known as the A quiver. This is shown in Figure 1.Figure 1: The A quiver.The total Lagrangian then consists of the four terms (see e.g. [21, 6]) S = S YM + S CS + S matter + S potential , (2.2)where the bosonic parts of the Chern-Simons and matter Lagrangian are S CS = X I =1 k I π Z Tr (cid:18) A I ∧ d A I + 23 A I ∧ A I ∧ A I + 2 D I σ I (cid:19) , (2.3) S matter = X a Z d x D µ ¯ φ a D µ φ a − ¯ φ a σ φ a + ¯ φ a Dφ a , (2.4)respectively, where φ a = ( A i , B i , Φ I ). In (2.4), the σ and D fields act in the appropriaterepresentation on the φ a – see [21, 6]. The Yang-Mills terms will, at low energies, be5rrelevant. Finally, the F-term potential is S potential = − X a Z d x (cid:12)(cid:12)(cid:12)(cid:12) ∂W∂φ a (cid:12)(cid:12)(cid:12)(cid:12) , (2.5)and we take the following superpotential: W = Tr (cid:2) s (cid:0) ( − n Φ n +11 + Φ n +12 (cid:1) + Φ ( A B + A B ) + Φ ( B A + B A ) (cid:3) . (2.6)Here n ∈ N is a positive integer, and s is a complex coupling constant. The super-potential is manifestly invariant under an SU (2) r flavour symmetry under which theadjoints Φ I are singlets and both pairs of bifundamentals A i , B i transform as doublets.There is also a Z flip2 symmetry which exchanges Φ ↔ Φ , A i ↔ B i , s ↔ ( − n s .The case n = 1 is special, since then the first two terms in (2.6) give a mass to theadjoint fields Φ , Φ . At low energy, we may therefore integrate out these fields. Onsetting s = k/ π , one recovers the ABJM theory with quartic superpotential [4] W ABJM = 4 πk ( A B A B − A B A B ) . (2.7)This theory is in fact superconformal with enhanced manifest N = 6 supersymmetry.We shall discuss the IR properties of the n > We denote the ranks by N = N + l , N = N , and consider the vacuum moduli spaceof the theory U ( N + l ) k × U ( N ) − k . In general there are six F-term equations derivedfrom imposing vanishing of (2.5), which is d W = 0: B i Φ + Φ B i = 0 , Φ A i + A i Φ = 0 ,s ( n + 1)Φ n + ( A B + A B ) = 0 ,s ( − n ( n + 1)Φ n + ( B A + B A ) = 0 . (2.8)One must also impose the three-dimensional analogue of the D-term equations [6], anddivide by the gauge symmetry. The reason for the subscript r will become apparent later. It is not to be confused with anR-symmetry.
6t is easier to understand this moduli space in stages, starting with the Abelian theorywith k = 1. In the U (1) × U (1) gauge theory, as usual in quiver theories the diagonal U (1) decouples (no matter field is charged under it). Precisely as in the ABJM theoryat Chern-Simons level k = 1, the anti-diagonal U (1), which we denote U (1) b , may begauged away because of the Chern-Simons interaction. Thus the vacuum moduli space,in the Abelian case with k = 1, is described purely by the set of F-terms (2.8). Thefirst four equations are reducible: either Φ = − Φ , or else A i = B j = 0 for all i, j .In the latter case the last two equations imply Φ = Φ = 0, so this is not a separatebranch. Thus Φ = − Φ holds in general, and we obtain the single equation for themoduli space s ( n + 1)Φ n + A B + A B = 0 . (2.9)After the change of coordinates z = ( A + B ), z = i2 ( A − B ), z = ( A + B ), z = i2 ( A − B ), z = ( s ( n + 1)) n Φ , this becomes simply X n ≡ ( z n + X a =1 z a = 0 ) . (2.10)For n = 1 this is indeed just C , as one expects since this is the Abelian ABJM theorywith k = 1, which corresponds to the theory on an M2-brane in flat spacetime. For n >
1, (2.10) instead describes an isolated four-fold hypersurface singularity, wherethe isolated singularity is at the origin { z = z = · · · = z = 0 } . This is Calabi-Yau, in the sense that away from the singular point there is a global nowhere-zeroholomorphic (4 , X , or X n when wewish to emphasize the n -dependence. In particular, X ∼ = C . We shall study thesevarieties in more detail later.The effect of changing the Chern-Simons levels to ( k, − k ) leads to a discrete quotientof the above vacuum moduli space by Z k ⊂ U (1) b [4, 22, 6]. Here by definition thecharges of ( A , A , B , B ) under U (1) b are (1 , , − , − k the Abelian vacuum moduli space is X n / Z k , where Z k actsfreely away from the isolated singular point. Thus X n / Z k is also an isolated four-foldsingularity.Having understood the moduli space for the U (1) k × U (1) − k theory, we may now turnto the general non-Abelian U ( N + l ) k × U ( N ) − k theory. The discussion here is similarto that for the ABJM theory in [4, 23]. In vacuum, Φ , σ are ( N + l ) × ( N + l ) matrices(with σ I Hermitian), Φ , σ are N × N matrices, while the A i and B i are N × ( N + l ) and7 N + l ) × N matrices, respectively. Note that using the gauge symmetry one may alwaysdiagonalize the σ I . The latter are fixed by the chiral field VEVs via three-dimensionalanalogues of the four-dimensional D-term equations [6], with the σ I playing the roleof moment map levels. If we take all matrices to be diagonal in the obvious N × N sub-blocks, so that the chiral fields take the form φ ABa = δ AB φ Aa , A, B = 1 , . . . , N , (2.11)with all other entries zero, then it is simple to see that the scalar potential is zeroprovided the φ Aa , A = 1 , . . . , N , satisfy the Abelian equations (the F-terms Φ A = − Φ A ,(2.9), and the D-term equations involving the σ AI ). It is also straightforward to see fromthe D-term potential that for generic σ I (meaning pairwise non-equal eigenvalues), alloff-diagonal fluctuations about any vacuum in this space of vacua are massive, withthe exception of fluctuations of Φ in the l × l sub-block. The diagonal ansatz forthe fields breaks the gauge symmetry to U (1) N × U ( l ) × U (1) N × S N , i.e. we obtainprecisely N copies of the Abelian N = 1 theory, where the permutation group S N permutes the diagonal elements (it is the Weyl group of the diagonal U ( N )). We alsoobtain a U ( l ) k Chern-Simons theory, as in [23], but for general n we also obtain asuperpotential term Ψ n +1 , where Ψ is an adjoint under U ( l ) coming from the l × l sub-block of Φ . Classically this has a trivial moduli space, since the F-term givesΨ = 0. Thus classically we obtain the symmetric product of N copies of the Abelianvacuum moduli space, i.e. Sym N ( X n / Z k ).However, as for the ABJM theory, in the quantum theory this moduli space can belifted. In particular, the U ( l ) k Chern-Simons theory with an adjoint superpotentialΨ n +1 has been studied in the literature before – for a recent account, together witha D-brane engineering of this theory, see for example [24] and [25]. As reviewed inthe latter reference, around equation (2.4), the above Chern-Simons theory has nosupersymmetric vacuum unless 0 ≤ l ≤ nk . This suggests that the above classicalspace of vacua is lifted unless this condition on l is obeyed. As we shall see later inthe paper, this condition is also realized non-trivially in the M-theory dual, and leadsto a 1-1 matching between the field theories U ( N + l ) k × U ( N ) − k , with 0 ≤ l < nk ,and the M-theory backgrounds we shall describe in section 3 (the theories with l = 0and l = nk will turn out to be dual to each other under a Seiberg-like duality that wederive using the Type IIB brane dual in section 4).8 .3 IR fixed points As mentioned already, for n = 1 the fields Φ , Φ are massive and on integrating theseout we recover at low energies the ABJM theory. This has N = 6 superconformalinvariance for general k ∈ Z . For n > n = 2 and equal ranks N = N = N flows to a strongly coupled N = 2 superconformal fixed point in theIR. The reason for this is that in this case there exists a candidate gravity dual: anAdS × Y / Z k Freund-Rubin solution of eleven-dimensional supergravity, where Y is aSasaki-Einstein seven-manifold. More precisely, the four-fold hypersurface singularity X admits a conical Calabi-Yau (Ricci-flat K¨ahler) metric, where the base of the coneis described by a homogeneous Sasaki-Einstein metric on Y – we shall discuss this indetail in section 3. Notice that, since W has R-charge/scaling dimension precisely 2,all of the fields φ a = ( A i , B i , Φ I ) must have R-charge/scaling dimension 2/3 at thisfixed point, showing that it is strongly coupled. As we shall also see in section 3, moreprecisely we conjecture this fixed point with equal ranks N to be dual to the Freund-Rubin Sasaki-Einstein background with zero internal G -flux: as for the ABJM theory[23], more generally it is possible to turn on l units of discrete torsion G -flux, wherein the gravity solution l is an integer mod nk , which is dual to changing the ranks to U ( N + l ) k × U ( N ) − k , as discussed at the end of the previous subsection.On the other hand, it was shown in [12] that for n > X n , for n >
2, do not have Calabi-Yau cone metrics. This indicates that the corre-sponding field theories cannot flow to conformal fixed points dual to these geometries.Indeed, the field theory realization of this was also described in [12]: if the superpo-tential is (2.6) at the IR fixed point, then the gauge invariant chiral primary operatorsTr Φ I have R-charge/scaling dimension 2 / ( n + 1); but for n > ≥ /
2, with equality only for a free field. It is thereforenatural to conjecture that for n > I in (2.6) are irrelevantin the IR, and thus s = 0 at the IR fixed point. If this is the case, then all the theorieswith n > same fixed point theory, namely the theory with s = 0.Consider then setting s = 0 in W in (2.6). If we also set k = 0, so that there isno Chern-Simons interaction, this is precisely the A quiver gauge theory. For equalranks N = N = N , the latter is well-known to be the low-energy effective theory9n N D2-branes transverse to R × C × C / Z ; here C / Z , where the generator of Z acts via ( z , z ) ( − z , − z ), is precisely the A singularity. The latter has anisolated singularity at the origin, where the N D2-branes are placed. This may beresolved by blowing up to O ( − → CP (the Eguchi-Hanson manifold). If we wrap l space-filling D4-branes over the CP zero-section, the ranks are instead N = N + l , N = N . This theory has enhanced N = 4 supersymmetry. If we now turn on theChern-Simons coupling k = 0, the Abelian vacuum moduli space of the resulting theoryis easily checked to be C × Con / Z k , where Con = { xy = uv } ⊂ C denotes the conifoldthree-fold singularity. Since this (non-isolated) four-fold singularity certainly admits aCalabi-Yau cone metric, this describes the candidate AdS dual to the IR fixed pointsof the theories with n >
2. It would be interesting to study this further. d = 4 , N = 1 theories and Laufer’s resolution As discussed in [6], the gauge group, matter content and superpotential of a d = 3, N = 2 Chern-Simons matter theory also specify a d = 4, N = 1 gauge theory – onetakes the same Yang-Mills action, matter kinetic terms and superpotential interaction,now defined in d = 4, and simply discards the Chern-Simons level data (since theChern-Simons interaction doesn’t exist in four dimensions). This is commonly referredto as the “parent theory”. The classical vacuum moduli space of this d = 4 parenttheory is closely related to that of the d = 3 Chern-Simons theory [6]. The stringtheoretic relation between the two theories was recently elucidated in [8], and we shallmake use of this correspondence later in the paper. The d = 4 parents of the abovetheories have been discussed extensively in the literature – in particular, see [26]. Weare not interested in the four-dimensional theories directly; however, it will be usefulto analyse their Abelian vacuum moduli spaces, and in particular the moduli spaceswith a non-zero Fayet-Iliopoulos (FI) parameter turned on.Compared to the d = 3 Chern-Simons matter theory, the only difference in con-structing the Abelian vacuum moduli space of the d = 4 parent is that the U (1) b gaugesymmetry now acts faithfully on the vacuum moduli space. The analysis of the F-termequations is identical to that in section 2.2, and for the Abelian theory with equalranks N = N = 1 we obtain the hypersurface equation (2.9). However, we must alsoimpose the D-term | A | + | A | − | B | − | B | = ζ , (2.12)and divide by U (1) b . Here we have introduced an FI parameter ζ ∈ R for U (1) b .10et us first set ζ = 0. In this case, the combination of the D-term (2.12) and iden-tifying by U (1) b may be realized holomorphically by taking the holomorphic quotientby the complexification C ∗ b . The charges of ( A , A , B , B ) are (1 , , − , − x = A B , y = A B , u = A B , v = A B . These satisfy the single relation xy = uv , (2.13)which is the conifold singularity. We must also impose the F-term (2.9), which setting z = ( s ( n + 1)) n Φ , as before, reads x + y + z n = 0 . (2.14)Combining (2.14) with (2.13), and again changing variables u = A B = i w − w , v = A B = i w + w , y = A B = i w − w n , z = [ s ( n + 1)] /n Φ = 2 /n w gives thethree-fold singularity W n ≡ (cid:8) w n + w + w + w = 0 (cid:9) . (2.15)This is an isolated three-fold singularity, and is again Calabi-Yau in the sense thatthere is a holomorphic volume form on the complement of the singular point { w = w = w = w = 0 } .Taking the parameter ζ = 0 in (2.12), one obtains a “small” resolution of the singular-ity W n . It is small in the sense that the singular point is replaced by a one-dimensional(rather than two-dimensional) complex submanifold – specifically, a CP . More pre-cisely, for ζ > W ζn ∼ = W + n , where “ ∼ =” means biholomorphic,while for ζ < W ζn ∼ = W − n . In both cases the “exceptional” CP has size | ζ | in the induced K¨ahler metric. Indeed, any K¨ahler metric on W ζn will have aK¨ahler class in H ( W ζn , R ) ∼ = R , and we regard ζ as specifying this K¨ahler class. Bothresolutions are also Calabi-Yau, in the sense that there is a holomorphic volume form,and are thus “crepant”. More on W ζn The end of this section is more technical, and may be skipped on a first reading.To see why W ζn takes the form described above, recall that the F-term equation(2.9) describes the moduli space in terms of coordinates ( A , A , B , B , Φ ) on C .Imposing the D-term (2.12) and dividing by U (1) b then gives Con ζ × C , where the11esolved conifold Con ζ is obtained from the quotient of the ( A , A , B , B ) coordinates,while the VEV of Φ is a coordinate on C . In particular, ζ > ζ < CP in the resolved conifold is at B = B = 0 for ζ >
0, and A = A = 0 for ζ <
0, respectively. The three-fold W ζn is then embedded in Con ζ × C via (2.9). We may also realize the D-term mod U (1) b as a C ∗ b quotient. Strictly speaking, this is a geometric invariant theory quotient, andfor ζ > { A = A = 0 } , while for ζ < { B = B = 0 } . Without loss of generality we henceforth take ζ > ζ < { A = A = 0 } from C , spanned by( A , A , B , B ). Define coordinate patches U i = { A i = 0 } ⊂ C , i = 1 ,
2. These willcover the manifold, as A and A cannot both be zero. On U the invariant functionsunder C ∗ b are spanned by x = A B , y = A B , u = A B , v = A B , ξ = A /A ,while on U the invariant functions are the same x, y, u, v , but instead µ = A /A . Wethen have the relations x = uξ , v = yξ , on U ,u = xµ , y = vµ , on U . (2.16)It follows that we may coordinatize U by ( u, y, ξ ) and U by ( x, v, µ ), with transitionfunctions ( x, v, µ ) = ( uξ, yξ, /ξ ) on the overlap U ∩ U . This shows explicitly theresolved conifold as O ( − ⊕ O ( − → CP , where ξ and µ are coordinates on the twopatches of the Riemann sphere CP , with µ = 1 /ξ on the overlap. The poles of thesphere are thus µ = 0 and ξ = 0.The three-fold W + n ∼ = W ζ> is embedded as a complex hypersurface in the resolvedconifold times C . We thus introduce patches H , with coordinates ( u, y, ξ, Z ), and H , with coordinates ( x, v, µ, Z ), where Z = Z = Φ is the coordinate on C . Theembedding equation (2.9) is then simply y = − uξ − Z n on H ,x = − vµ − Z n on H . (2.17)We may thus eliminate x and y and coordinatize H by ( u, ξ, Z ) and H by ( v, µ, Z ),with transition functions ( v, µ, Z ) = ( − ξZ n − ξ u, /ξ, Z ) on the overlap H ∩ H .This is precisely the description of the small crepant resolution W + n of W n given byLaufer [27]. One sees explicitly the exceptional CP with coordinates ξ, µ , and µ =1 /ξ on the overlap. One also sees that for n = 1 the normal bundle of CP inside12 + n is O ( − ⊕ O ( − → CP , while for all n ≥ O (0) ⊕ O ( − → CP . In this section we discuss M-theory and Type IIA duals to the Chern-Simons-quivertheories of section 2.1. We have already shown that the vacuum moduli space ofthe U ( N + l ) k × U ( N ) − k theory is Sym N X n / Z k , and this suggests a dual M-theoryinterpretation in terms of N M2-branes probing the four-fold singularity X n / Z k . As in[23], we show that the integer l , which is constrained to lie in the interval 0 ≤ l ≤ nk in the field theory, may be identified with turning on l units of torsion G -flux in theM-theory background. On the gravity side, l is defined only modulo nk – we will haveto wait until section 4 to see why the l = 0 field theory is dual to the l = nk theory.As already mentioned, only for n = 1, n = 2 do the four-fold singularities X n haveRicci-flat K¨ahler cone metrics, implying that only in this case do the conformal fixedpoints of the Chern-Simons-quiver theories have AdS duals of this type; we conjecturedthat for all n > same fixed point theory in the IR, and thatthis has a different AdS dual description where the Sasaki-Einstein seven-space is thesingular link of C × Con / Z k . Although we are interested primarily in the case n = 2,we retain n throughout this section and study M-theory on AdS × Y n / Z k , where Y n is the link of the singularity X n . We stress again, however, that the AdS solutions ofthis type exist only for n = 1, n = 2. The discussion of section 2.2 suggests that the Chern-Simons quivers of section 2.1should have M-theory duals in terms of M2-branes placed at the four-fold singularities X n / Z k (2.10). Thus it is natural to conjecture that the IR fixed points of the Chern-Simons quivers, for n = 1, n = 2, are SCFTs dual to the gravity backgrounds AdS × Y n / Z k , where Y n is the base of the cone X n , equipped with a Sasaki-Einstein metric.The case n = 1 is just the round metric on Y = S , which is the ABJM model. Thecase n = 2 leads instead to Y = V , , where V , has a homogeneous Sasaki-Einsteinmetric that we discuss below.Consider the complex cone X n defined in (2.10). We may define the compact seven-13anifold Y n via Y n ≡ X n ∩ S , (3.18)where S = { P i =0 | z i | = 1 } ⊂ C . For n = 1 this is simply Y = S , so we focus ondescribing Y . In this case X is a complex quadric, and the vector action of SO (5)on the coordinates z i acts transitively on the seven-manifold Y , and thus Y = V , = SO (5) /SO (3) is a coset space. X is also invariant under the rescaling z i λz i , for λ ∈ C ∗ , and the quotient B ≡ ( X \{ } ) / C ∗ is a compact complex manifold of complexdimension three. Equivalently, this may be defined as B = V , /U (1) R , where U (1) R acts on the z i with charge 1, and thus B ∼ = Gr , = SO (5) /SO (3) × SO (2) is also acoset space. The space Gr , is the Grassmanian of two-planes in R .There is an explicit homogeneous Sasaki-Einstein metric on Y = V , , so that thequadric singularity X has a Ricci-flat K¨ahler cone metric. The Reeb U (1) actionis precisely the action by U (1) R ⊂ C ∗ above; thus V , is a regular Sasaki-Einsteinmanifold and the quotient Gr , is a homogeneous K¨ahler-Einstein manifold. TheSasaki-Einstein metric on V , may be written explicitly in suitable coordinates [28]d s ( V , ) = 916 (cid:20) d ψ + 12 cos α (d β − cos θ d φ − cos θ d φ ) (cid:21) + d s (Gr , ) , (3.19)where d s (Gr , ) = 332 h α + sin α (d β − cos θ d φ − cos θ d φ ) + (1 + cos α )(d θ + sin θ d φ + d θ + sin θ d φ )+ 2 sin α cos β sin θ sin θ d φ dφ − α cos β d θ d θ + 2 sin α sin β (sin θ d φ d θ + sin θ d φ d θ ) i (3.20)is the homogeneous K¨ahler-Einstein metric on B = Gr , . The ranges of the coordi-nates are0 ≤ θ i ≤ π , ≤ φ i < π , ≤ ψ < π , ≤ α ≤ π , ≤ β < π . (3.21)The volume of the Sasaki-Einstein metric on V , is [28]vol( V , ) = 27128 π . (3.22)Notice the isometry group of the homogeneous metric on V , is SO (5) × U (1) R , andthus in particular this is a non toric manifold.14hus for n = 1, n = 2 we have supersymmetric Freund-Rubin backgrounds of eleven-dimensional supergravity of the type AdS × Y n , with Y = S and Y = V , . Themetric and G -field take the form d s = R (cid:18)
14 d s (AdS ) + d s ( Y n ) (cid:19) ,G = 38 R dvol(AdS ) . (3.23)The AdS radius R is determined by the quantization of the G -flux N = 1(2 πl p ) Z Y n ∗ G , (3.24)where l p is the eleven-dimensional Planck length, given by R = (2 πl p ) N Y n ) . (3.25)We also note that vol( Y = S ) = π / U (1) b .Writing the complex cone as X n = { z n + A B + A B = 0 } , the U (1) b symmetry actson ( z , A , A , B , B ) with charges (0 , , , − , − Y n defined in (3.18), and it is easy to see that this is a free action, i.e. there are nofixed points on Y n . For both n = 1, n = 2, U (1) b acts isometrically on the Sasaki-Einstein metrics. In particular, for n = 2 this embeds into the isometry group as U (1) b ∼ = SO (2) diagonal ⊂ SO (4) ⊂ SO (5). This is a non-R isometry, and so preservesthe Killing spinors on Y = V , . We may thus take a quotient of V , by Z k ⊂ U (1) b to obtain a Sasaki-Einstein manifold V , / Z k with π ( V , / Z k ) ∼ = Z k . Since SO (4) ∼ =( SU (2) l × SU (2) r ) / Z , the diagonal SO (2) in SO (4) is U (1) b ∼ = U (1) l ⊂ SU (2) l . Thusthe isometry group of the quotient space V , / Z k is SU (2) r × U (1) b × U (1) R . This isthe manifest global symmetry in the Chern-Simons-quiver theories.We conjecture that the Chern-Simons-quiver theory U ( N ) k × U ( N ) − k , with mattercontent given by the quiver in Figure 1 and superpotential interaction (2.6) with n = 2,flows to a conformal fixed point in the IR, and is dual to the above AdS × Y / Z k M-theory background. As evidence for this, we have shown that the moduli spaceof the field theory agrees with the moduli space of N M2-branes probing the conegeometry, and that the isometry group of the AdS solution precisely matches the The Einstein metrics on AdS and Y n obey Ric AdS = − g AdS , Ric Y n = 6 g Y n , respectively. of the field theory. Later in sections 3.3 and 3.4 we shall present amatching of various gauge invariant chiral primary operators to supergravity multipletsand certain supersymmetric wrapped D-branes, respectively, as further evidence. Insection 4 we will also present a Type IIB brane construction.Let us now discuss turning on a torsion C -field, corresponding to the addition offractional branes [23]. As shown in appendix A, in general we have H ( Y n / Z k , Z ) ∼ = Z nk , and thus we may turn on a torsion G -field, i.e. a flat, but topologically non-trivial, G -flux. Each different choice of such G -flux will lead to a physically distinct M-theory background. We may equivalently describe this as a (discrete) holonomy for thethree-form potential C through the Poincar´e dual generator Σ of H ( Y n / Z k , Z ) ∼ = Z nk .Thus 1(2 πl p ) Z Σ C = lnk mod 1 . (3.26)Since the physical gauge invariant object is a holonomy, the integer l above is onlydefined modulo nk . Equivalently, this labels the G -flux [ G ] = l ∈ H ( Y n / Z k , Z ) ∼ = Z nk .For each choice of l with 0 ≤ l < nk we therefore have a 1-1 matching of the M-theorybackgrounds to the field theories with gauge groups U ( N + l ) k × U ( N ) − k . We shallpresent further evidence for matching the G -flux to the ranks in this way from theType IIA dual in section 3.5. When k ≫ N ≫ k the radius of the U (1) b circle becomes small and a better descriptionis obtained by reducing the background along U (1) b to a Type IIA configuration. Since U (1) b acts freely on Y n , we may define quite generally M n = Y n /U (1) b , which is asmooth six-manifold. For n = 1 this gives M = CP , while for n > M n has the same cohomology groups as CP , but a cohomology ring that depends on n , as shown in appendix A. For n = 2, U (1) b is a non-R symmetry, and therefore allsupersymmetries are preserved in the quotient V , /U (1) b = M . On the other hand,the Type IIA reduction of N = 2 Freund-Rubin backgrounds along the R-symmetry(Reeb vector) direction breaks supersymmetry [30]. In particular, we stress that M As often happens in AdS / CFT , for k = 1 the isometry group is enhanced. In particular we have SO (5) × U (1) R symmetry, rather than the SU (2) r × U (1) b × U (1) R symmetry valid for k >
1. Thisformer symmetry is not manifest in the UV Lagrangian. It is important here that the G -flux is classified topologically by H ( Y, Z ), which is true only ifthe membrane anomaly is zero [29]. In fact the membrane anomaly always vanishes on any orientedspin seven-manifold. different from the K¨ahler-Einstein six-manifold Gr , = V , /U (1) R introduced insection 3.1. These types of reduction were discussed in [31], and we now recall theiressential features.To perform the reduction we write the Sasaki-Einstein metric on Y n / Z k asd s ( Y n / Z k ) = d s ( M n ) + wk (d γ + kP ) , (3.27)where γ has 2 π period. We then obtain the following Type IIA string-frame metricand fields d s = √ w R k (cid:18)
14 d s (AdS ) + d s ( M n ) (cid:19) , (3.28)e = R k w / , F = 38 R dvol(AdS ) , F = kl s g s d P , (3.29)where w is a nowhere-zero bounded function on M n (since U (1) b acts freely). The RRtwo-form flux has quantized periods, namely12 πl s g s Z Σ F = k . (3.30)Here Σ ⊂ M n is the generator of H ( M n , Z ) ∼ = Z . Of course, these supergravitysolutions exist only for n = 1, n = 2. In the latter case, then more precisely in termsof the coordinates in (3.19), (3.20) we have that γ = φ and w = 332 (cid:2) cos α (1 + sin θ ) (cid:3) . (3.31)The torsion C -field reduces to a flat NS B -field in Type IIA [23] via C = A + B ∧ d ψ . (3.32)Here A denotes the RR three-form potential, while ψ parametrizes the M-theory circlewith period 2 πl s g s , where recall that l p = l s g / s is the eleven-dimensional Planck length.Denoting with Ω = [d P/ π ] the generator of H ( M n , Z ) ∼ = Z , we then have B = (2 πl s ) lkn Ω . (3.33) A detailed discussion of the topology of M n is contained in appendix A. The authors of [32] argue, for the ABJM theory n = 1, that there is a shift in this B -fieldperiod by 1 / πl s ) ). Notice that, ordinarily, the B -field period through Σ would bea modulus, able to take any value in S (after taking account of large gauge transformations). Sincethis does not affect our discussion, we shall not study this further here. B through Σ is hence b ≡ πl s ) Z Σ B = lkn mod 1 . (3.34)Again, as for the C -field period (3.26) through Σ , this is only defined modulo 1. InType IIA, this is because large gauge transformations of the B -field change the period b by an integer. We now turn to a discussion of the chiral primary operators of the N = 2 gauge theorywith n = 2, and how they are realized in the gravity dual. In the field theory we can con-struct chiral primary operators by taking appropriately symmetrized gauge-invarianttraces of products of fields. These operators may be denoted very schematically asTr [Φ n ( AB ) n ]. They are invariant under U (1) b , and their dimension at the n = 2 IRfixed point is ∆ = 2 / · ( n + 2 n ). However, because of the presence of monopole oper-ators in three dimensions, these do not exhaust the list of all chiral primaries [4]. Themonopole operator with a single unit of magnetic flux in the diagonal U (1) transformsin the (Sym k ( N ) , Sym k ( ¯ N )) representation of the gauge group, and following [4] wemay denote it as e i τ . Using this we can construct generalized gauge-invariant traces asTr [Φ n ( AB ) n A m k B m k e i( m − m ) τ ] , n i , m i ∈ N . (3.35)It is currently not known how to compute the dimensions of monopole operators instrongly coupled N = 2 Chern-Simons theories [33]. However, it is plausible that inthe present case, as conjectured for the ABJM theory [4], their scaling dimension iszero. Assuming this, the dimensions of the operators (3.35) are then∆ = 23 [ n + 2 n + ( m + m ) k ] . (3.36)These operators may be matched to a tower of states in the Kaluza-Klein spectrumon V , derived in [11]. Consider first setting k = 1. The spectrum is arranged intosupermultiplets, labelled by representations of Osp (4 | × SO (5) × U (1) R . When thecorresponding dimensions of dual operators are rational, the multiplets undergo short-ening conditions [34]. In particular, we see from Table 6 of [11] that a certain vectormultiplet (“Vector Multiplet II”) becomes a short chiral multiplet , with componentsdenoted as ( S/ Σ , λ L , π ). These have spins (0 + , / , − ), respectively, and dimensions18∆ , ∆ + 1 / , ∆ + 1), with ∆ = 23 m , m = 1 , , . . . . (3.37)The lowest component fields then match the operators (3.35) with m = n + 2 n + m + m .For k > Z k projection . This is mosteasily seen using the equivalence of chiral primary harmonics on V , to holomorphicfunctions on the Calabi-Yau cone singularity X [12]. These can be expanded inmonomials of the form Q i =0 z s i i , for s i ∈ N . Using the results of [12] (see equation (3.22)of this reference) we determine that the R-charges associated to the coordinates z i are all equal to 2 /
3, which of course agrees with (3.37). When k > z + A B + A B = 0 , (3.38)which diagonalizes the action of Z k ⊂ U (1) b . Recall that under U (1) b these coordinateshave charges (0 , , , − , − k > z n A p B p , p − p = 0 mod k , p i ∈ N . (3.39)These of course match precisley with the operators (3.35), where p = n + m k , p = n + m k .For later purposes it will be useful to discuss the structure of the chiral multiplets onthe gravity side in a little more detail. The lowest bosonic components S/ Σ arise froma linear combination of metric modes and C -field modes in AdS . The top bosoniccomponents π come purely from C -field modes in the internal directions, namely fromcertain massive harmonic three-forms on Y = V , – see Table 1 of [11].In the field theory, a chiral superfield may be written in superspace notation asΦ = φ + θψ + θ F . The component fields have R-charges (∆ , ∆ − , ∆ −
2) and scalingdimensions (∆ , ∆ + 1 / , ∆ + 1), respectively. Then the bosonic physical degrees offreedom of a chiral operator of the form Tr Φ m are a scalar φ m with dimension m ∆, The representations that survive the Z k projection are the singlets in the decomposition of [ m, SO (5) → SU (2) r × U (1) b . For general n , the would-be R-charges are n/ ( n + 1) for the coordinates z , . . . z and 2 / ( n + 1)for the coordinate z . Therefore for n > ≥ /
2, whichgeometrically is the Lichnerowicz bound. For n = 3 it saturates this bound, but one can still arguethat the corresponding Sasaki-Einstein metric on Y does not exist [12]. ψ α ψ α φ m − with dimension m ∆ + 1. In the gravity dual, these aredual to the scalar modes S/ Σ and the pseudoscalar modes π , respectively. In this section we briefly discuss M5-branes wrapped on certain supersymmetric sub-manifolds in Y n / Z k , and their Type IIA incarnation as D4-branes wrapped on subman-ifolds in M n . These correspond to certain “baryonic” ( i.e. determinant-like) operatorsin the field theories.A full analysis of the spectrum of baryon-type operators is beyond the scope of thispaper. However, we may provide further evidence for the proposed duality by analysinga certain simple set of operators. Thus, for the adjoint fields Φ I we may consider thegauge-invariants det Φ I , I = 1 ,
2. Notice that Φ is an ( N + l ) × ( N + l ) matrix, whileΦ is N × N . We may also define the (in general non -gauge-invariant) operators A γ ··· γ l i ≡ N ! ǫ α ··· α N A α i β · · · A α N i β N ǫ β ··· β N γ ··· γ l , B i γ ··· γ l ≡ N ! ǫ α ··· α N B β i α · · · B β N i α N ǫ β ··· β N γ ··· γ l . (3.40)Here A i lives in Λ l ( N + l ), the l th antisymmetric product of the anti-fundamentalrepresentation of U ( N + l ), while B i lives in Λ l ( N + l ) [35]. These are gauge-invariantonly for l = 0, but even in this case one needs to insert an appropriate monopoleoperator (see [33, 36] for a recent discussion of these operators); we will not studythis here. For l >
0, one can obtain gauge-invariant operators by, for example, taking( N + l ) copies of A i and then contracting with l epsilon symbols for U ( N + l ) (withappropriate monopole operators). This situation is clearly much more complicated thanit is for D3-branes in Type IIB string theory, and deserves further study. However, asfor the ABJM theory, the operators (3.40) can still be matched to wrapped branes inthe gravity dual, as we shall explain.In M-theory we may associate these types of operators to M5-branes wrapping su-persymmetric submanifolds. More precisely, these are the boundaries of divisors inthe Calabi-Yau cone – see, e.g. , the first reference in [9]. Given the discussion of theAbelian moduli space in section 2.2, we may associate the operators det Φ I with thedivisor { z = 0 } in the Calabi-Yau cone, while A is associated to { z = i z } , A to { z = i z } , B to { z = − i z } , and B to { z = − i z } . This follows by noting that,in the Abelian theory, the operators may be regarded as sections of line bundles over20he Abelian vacuum moduli space; the divisors we have written are then the zeros ofthese sections.Let us consider first the adjoints. Setting z = 0 in X n gives { z + z + z + z = 0 } ,which is a copy of the conifold singularity. Thus the boundary Σ (0) n of this divisor is acopy of T , , for all n . Taking the Z k quotient, one obtains instead Σ (0) n / Z k = T , / Z k ,where recall that Z k is embedded in the diagonal SO (2) in SO (4). For the main caseof interest, n = 2, this can be seen explicitly in the polar coordinates of section 3.1: thefive-dimensional submanifold Σ (0)2 corresponds to setting α = β = 0, and its volume isvol(Σ (0)2 ) = (3 π ) / . We may also compute this volume using the results of [28, 12].This gives the general result vol(Σ (0) n ) = ( n + 1) π n . (3.41)This is the volume of the submanifold induced by any Sasakian metric on Y n withReeb vector field weights (4 / ( n + 1) , n/ ( n + 1) , n/ ( n + 1) , n/ ( n + 1) , n/ ( n + 1)).The latter are normalized so that the holomorphic (4 , Y n ) = ( n + 1) π n . (3.42)This is then the volume of a Sasaki-Einstein metric on Y n if it exists , which is true onlyfor n = 1, n = 2. Using the formula for the dimension of the dual operator [37]∆ = N π vol(Σ)vol( Y ) , (3.43)we obtain in general ∆[det Φ I ] = 2 N/ ( n + 1). Notice here that, since Σ (0) n is invariantunder U (1) b , after taking the Z k quotient the dependence on k in the numerator anddenominator in (3.43) cancel. This result then matches with the conformal dimen-sions of the adjoints computed from the constraint that the superpotential has scalingdimension 2.However, the above discussion overlooks an important subtlety: we have two op-erators det Φ , det Φ , but only one divisor. Moreover, in the case of unequal ranks, U ( N + l ) k × U ( N ) − k , one expects det Φ to have dimension ∆ ∝ N + l , while det Φ should have dimension ∆ ∝ N . In the case of D3-branes wrapping supersymmetricthree-submanifolds in Sasaki-Einstein five-manifolds, there can also be multiple bary-onic operators mapping to the same divisor: they are distinguished [38] physicallyin the gravity dual by having different flat worldvolume connections on the wrapped213-branes. Here we have a wrapped M5-brane, and thus one expects the self-dualtwo-form on its worldvolume to play a similar role. Notice also that in general in theconformal dimension formula (3.43) one expects the on-shell M5-brane worldvolumeaction to appear in the numerator. In general this action depends on both the self-dual two-form and the pull-back of the C -field, reducing simply to the volume of Σwhen both are zero. Of course, l = 0 corresponds in the gravity dual to having a non-zero flat C -field. Similarly, in the Type IIA dual picture that we discuss below theseare wrapped D4-branes, whose conformal dimensions should be related to the on-shellDirac-Born-Infeld action, including the B -field (3.33). We shall not investigate thisfurther here, but instead leave it for future work.The remaining four dibaryon operators in (3.40) correspond to the same type ofsubmanifold; hence, without loss of generality, we shall study the A operator. Thelocus { z = i z } in the Calabi-Yau cone X n cuts out a singular subvariety for general n : clearly, z may take any value in C , but the remaining defining equation of X n implies that z n + z + z = 0, which is a copy of the A n − singularity. Thus the divisorof interest is C × ( C / Z n ), and the intersection with Y n is then a copy of the singular space Σ (1) n = S / Z n . On the other hand, the Z k quotient acts freely on Σ (1) n . Thevolume may again be computed from the character formula [12], givingvol(Σ (1) n ) = ( n + 1) π n , (3.44)and hence conformal dimension ∆[ A i ] = nN/ ( n + 1). Again, notice this preciselymatches the scaling dimensions of the fields A i obtained by imposing that the super-potential has scaling dimension 2.It is instructive to also consider the reduction to Type IIA. The wrapped M5-branesabove then become D4-branes wrapped on four-dimensional subspaces Σ ( i ) n /U (1) b .Since the quotient by U (1) b does not break supersymmetry of the background, weexpect that the four-dimensional submanifolds here will also be supersymmetric; how-ever we have not checked the kappa-symmetry of the wrapped D4-branes explicitly.The reduction of Σ (0) n is diffeomorphic to S × S . More interesting is the reductionof the (singular) Σ (1) n subspaces, corresponding to the dibaryonic operators (3.40) with l uncontracted indices. The latter dependence on l may be understood by analysing acertain tadpole in Type IIA, as for the ABJM theory. To discuss the reduction to TypeIIA, it is more convenient to use the coordinates A i , B i . The divisor corresponding tothe A operator is then simply { z = i z } = { A = 0 } . The group U (1) b acts withcharge − B , and charges (1 , −
1) on ( A , B ). The A n − singularity22n these coordinates is z n + A B = 0. Denoting by u , u standard coordinates on C under which Z n acts as (e πi/n , e − πi/n ), then the invariant functions under Z n are A = u n , B = u n and z = e iπ/n u u , from which one sees explicitly that A B = − z n .Thus U (1) b acts with weights (1 /n, − /n ) on the coordinates ( u , u ). This impliesthat the quotient is topologically Σ (1) n /U (1) b = ( S / Z n ) /U (1) b ∼ = WCP n, , . Thelatter is the subspace on which the D4-brane is wrapped. It has an isolated Z n orbifoldsingularity at the image of A = B = 0, which lifts to the A n − singularity. Asimple topological description of WCP n, , is to take O ( n ) → CP , and then collapsethe boundary, which is S / Z n , to a point. The latter is then the isolated singularity.Conversely, the image of B = 0 is a smooth two-sphere which lifts to the S / Z n linkof the A n − singularity. Thus in general the integral of F / (2 πl s g s ) over this S in WCP n, , is equal to nk .Now, from appendix A we have that H ( M n , Z ) ∼ = Z . Call the generator Σ . It isalso shown in this appendix that the integral of the square of Ω = 1 ∈ H ( M n , Z ) ∼ = Z over Σ is equal to n . Now, in general also [ F / πl s g s ] = k Ω , and since the first Chernclass of O ( n ) → CP is n , it follows that the integral of the pull-back of Ω ∧ Ω over WCP n, , is equal to n /n = n . This implies that the copy of WCP n, , on whichthe BPS D4-brane is wrapped is a (singular) representative of the four-cycle Σ in thesmooth six-manifold M n .Consider now the Wess-Zumino couplings on the D4-brane wrapped on WCP n, , .Due to the presence of the B -field (3.33), we obtain the term1(2 π ) l s Z R time A · Z Σ B ∧ F = l · g s πl s Z R time A . (3.45)Here we have performed the calculation Z Σ lnk Ω ∧ k Ω = l . (3.46)The Wess-Zumino coupling thus induces a tadpole for the worldvolume gauge field A .To cancel this tadpole requires that l fundamental strings end on the D4-brane. Inthe field theory this corresponds to the fact that the dibaryon operators (3.40) haveprecisely l uncontracted indices [23]. This assumes that the worldvolume gauge field flux on Σ is zero. In fact for odd n , the smoothlocus of the wrapped submanifold Σ = WCP n, , is not spin, and thus one must turn on a 1 / / B (in the case n = 1) in footnote 9, which cancels this. In our case ofinterest, n = 2, there is no such shift. l is defined only modulo nk , while in the field theory 0 ≤ l ≤ nk . In particular,when one states that the tadpole requires l fundamental strings to end on the D4-brane,this is only true modulo nk . Thus, it must be that nk fundamental strings are physicallyequivalent to none. In fact this is easy to see in the M-theory lift. The strings lift to nk M2-branes ending on the M5-brane. More precisely, the end of the M2-branes wrapthe M-theory circle that is a smooth S in Σ (1) n , together with the time direction inAdS . If we remove the singular locus from Σ (1) n , which is a copy of S , we obtain asmooth manifold with fundmental group Z nk – removing the singular locus is sensible,since the supergravity approximation will break down near to this locus. This resultimplies that nk M2-branes ending on the M5-brane can “slip off”, since nk copies ofthe circle that they wrap are contractible on the M5-brane worldvolume. This matchesnicely with the fact that this is equivalent, via (3.45), to a large gauge transformationof the B -field. There is a different way of thinking about the Type IIA backgrounds discussed insection 3.2, which we explain in this section. This demonstrates rather directly the re-lationship with the “parent” four-dimensional field theories, and elucidates the stringyorigin of the Chern-Simons-quiver theories. We will also need the present discussionto derive a Type IIB Hanany-Witten-like brane configuration in the next section.We begin by considering the geometry R , × X n / Z k in M-theory, where X n is the conesingularity (2.10), together with N spacefilling M2-branes. The U (1) b circle acts freelyaway from the cone point, and thus we can reduce to a Type IIA geometry R , × C ( M n ),with k units of RR two-form flux through the generator of H ( M n , Z ) ∼ = Z . In thispicture we have N spacefilling D2-branes. However, we may instead take the K¨ahlerquotient of X n / Z k by U (1) b , at level ζ ∈ R , to obtain precisely the three-fold W ζn introduced in section 2.4. For ζ = 0, recall this is the affine three-fold given by (2.15),while for ζ = 0 one instead obtains Laufer’s small resolution of this singularity, whichhas a blown-up CP of size | ζ | . The latter is the Abelian vacuum moduli space ofthe four-dimensional parent theory, as discussed in section 2.4. This picture describesthe seven-dimensional space C ( M n ) as a fibration of W ζn over the real line R thatparametrizes the moment map level ζ , as shown in Figure 2.Indeed, we can instead consider starting with Type IIA on R , × R × W n , where24igure 2: The Type IIA reduction of M-theory on X/ Z k on U (1) b is C ( M n ). Thisgeometry may also be viewed as a fibration of W ζn over the R direction, where the size | ζ | of the exceptional CP depends on the position in R . In particular, the conicalsingularity of C ( M n ) is the conical singularity of W n above the origin in R . The aboveschematic picture would be precisely the toric diagram in the case n = 1 (for n > R = R for later convenience, with N spacefilling D2-branes. Here W n should of course be equipped with some kind of Calabi-Yau metric, although wenote that from [12] it does not admit a conical Calabi-Yau metric for n > n = 1is the conifold). We might imagine W n as modelling a local singularity in a compactCalabi-Yau manifold, in which case the Calabi-Yau metric here would in any casebe incomplete. If we now T-dualize along the (compactified) R direction, then weprecisely obtain the Type IIB string theory set-up yielding the four-dimensional parenttheory. We may also replace the singular three-fold by its crepant resolution W ζn ,thinking of ζ as parametrizing the period of the K¨ahler form through the exceptional CP . We may then turn on k units of RR two-form flux through this CP , although inorder to preserve supersymmetry it is necessary to also fibre the size of the CP over the R direction – this may be seen by appealing to the reduction of the M-theory solutionabove. Thus we identify R ∼ = { ζ ∈ R } . If µ b denotes the moment map for U (1) b , sothat µ b : X n / Z k → R , then notice that the inverse image of ζ ∈ R is µ − b ( ζ ) = W ζn ,so that in particular the cone geometry appears at the origin in R . By construction,the RR two-form flux may then be identified with the first Chern class c ∈ H ( W ζn , Z )25f the U (1) b M-theory circle bundle. One can then compute that12 πl s g s Z C P F = k . (3.47)As explained in [8], the above picture leads to a physical relation between the parenttheory and the Chern-Simons theory. If we have N spacefilling D2-branes togetherwith l fractional D4-branes wrapping the (collapsed) CP in W n , the resulting gaugetheory is precisely the A quiver theory with superpotential (2.6), with gauge group U ( N + l ) × U ( N ) – this is discussed, for example, in [26]. The key result in [8] isthat the addition of the k units of RR two-form flux through the CP then induces aChern-Simons interaction with levels ( k, − k ) for the two nodes, respectively, via theWess-Zumino terms on the fractional branes. This leads to a Type IIA string theory derivation of our Chern-Simons-quiver theories, starting with the geometric engineeringof the parent theory. Also notice that the l fractional D4-branes, wrapped on thecollapsed CP , will lift to l fractional M5-branes – since the M5-brane is a magneticsource for the G -field, it is thus natural to identify the l units of torsion G -flux withthe l fractional M5-branes. Indeed, more precisely, a copy of the exceptional CP at ζ > H ( M n , Z ) ∼ = Z , and this lifts to the generator Σ of H ( Y n / Z k , Z ) ∼ = Z nk , as shown in appendix A. Thus l fractional D4-branes wrappedon the CP lift to l fractional M5-branes wrapped on Σ . The latter is then Poincar´edual to l units of torsion G -flux. In this section we derive a Hanany-Witten-like brane configuration in Type IIB stringtheory. This takes the usual form of D3-branes (wrapped on a circle) suspended be-tween 5-branes, except that for n > n = 2 supergravity solution may wish to skip ahead to section 5. k = 0 We begin with the Type IIA background of R , × R × W ζn , with zero RR flux, discussedat the end of the previous section. Here we have included a K¨ahler class ζ ∈ R , whichis a free parameter, so that for ζ = 0 W ζn is a smooth non-compact K¨ahler manifold.26or ζ = 0, we are considering the singular three-fold W n . We rewrite the definingequation (2.15) as W n = { w n + w − uv = 0 } ⊂ C , (4.48)where as before u = i w − w , v = i w + w . We may then consider performing a T-duality along U (1) ≡ U (1) that acts with charge 1 on u and charge − v . We mayalso consider the K¨ahler quotient by U (1) , with moment map µ = | u | − | v | , whichmaps µ : W n → R ≡ R , where we have introduced the subscript 7 to distinguish thiscopy of R from R above. It follows that { C = h u, v i} //U (1) ∼ = C , for any value of µ , and hence similarly W n //U (1) ∼ = C . Indeed, the defining equation of W n is then w n + w = w , where w = uv is the coordinate on C = C / C ∗ . We may thus eliminatethe coordinate w to see that W n //U (1) ∼ = C , spanned by the coordinates w , w , forany value of the moment map. It follows that W n /U (1) is a C fibration over R , andthus W n /U (1) ∼ = R × C ∼ = R .There are, however, fixed points of U (1) . If we peform a T-duality along U (1) ,the above shows that the T-dual spacetime is R , × R × S × R × C , where S isthe U (1) circle after performing the T-duality. However, there are codimension fourfixed point sets of U (1) , where the action on the normal fibre is the standard Hopfaction on R . These become NS5-branes in the T-dual Type IIB picture. The fixedlocus here is u = v = 0, which is the origin in the moment map direction R . In the C direction they cut out the locus w n = − w in C , which is w = ± i w n . These are twocopies of C embedded as affine algebraic curves in C , which intersect over the origin { w = w = 0 } . Note that when n = 1, which is the ABJM case, we see w = ± i w are two linearly embedded copies of C . This is indeed the standard Hanany-Wittenbrane configuration for the conifold [39]. For n >
1, we obtain a non-linear version ofthis, where the NS5-branes are embedded as the curves w = ± i w n in C . We labelthe latter directions 4589, and refer to C . The NS5-branes also sit at a point in the S circle, where their distance of separation is the period of the B -field through thecollapsed CP in W n . The final Type IIB picture is described in Figure 3.Note we can immediately read off the matter content of the field theory from thispicture: the brane set-up is identical, apart from the embedding of the NS5-branes in4589, to the A singularity. Thus we may read off two gauge groups, corresponding tothe N D3-branes breaking on the two NS5-branes on the S circle. At each NS5-branewe obtain a pair of bifundamentals, A i , B i , and an adjoint Φ , Φ for each D3-branesegment. The A theory also has the N = 4 cubic superpotential for these fields. For27igure 3: The Type IIB brane dual of the Type IIA background R , × R × W n with N spacefilling D2-branes. The Type IIB spacetime is flat: R , × R × S × R × C .There are N D3-branes filling the R , directions and wrapping the S circle; theyare at the origin in R , R and C . There are two NS5-branes that are spacefillingin R , and separated by a distance in the S circle that is given by the period of B through the collapsed CP in the T-dual three-fold geometry W n ; they both sit atthe origin in R , fill the R direction, and wrap the holomorphic curves w = ± i w n ,respectively, in C with complex coordinates w , w . These curves intersect at theorigin w = w = 0. n = 1 is the standard Hanany-Witten brane configuration for theconifold singularity, where the NS5-branes are linearly embedded.the A theory, both branes are parallel, say at the origin in the 89 plane. For theconifold theory n = 1, one brane is in the 45 plane, while the other is in the orthogonal89 plane. This corresponds to giving a mass to the adjoints, -Φ + Φ , as shown in [39].Integrating these out, one obtains the quartic superpotential of Klebanov-Witten. Inthe general n case, the non-trivial embedding of the NS5-branes in C is reflected inthe higher order ( − n Φ n +11 + Φ n +12 superpotential term. k = 0 The next step is to turn back on the RR two-form flux, so that k = 0: this is then theType IIA dual of M-theory on X n / Z k with N spacefilling M2-branes. As we discussedin section 3.5, supersymmetry also requires that one fibre the parameter ζ over the R direction. Thus, before discussing this, we first consider the effect of turning on theparameter ζ in the T-dual IIB brane set-up above.Without loss of generality, we take ζ > W ζn ∼ = W + n is biholomorphic toLaufer’s resolved manifold, with an exceptional CP replacing the singular point of28 n . The U (1) action on W n extends to an action on W + n . To see this, recall fromthe last part of section 2.4 that ( A , A , B , B , z ) are coordinates on C , and that x = A B , y = A B , u = A B , v = A B are invariants under U (1) b , with ξ = A /A an invariant on U and µ = A /A an invariant on U . The embedding equation (2.9)then becomes x + y + z n = 0. When ζ = 0 we have the conifold xy = uv , andeliminating x this becomes y + yz n + uv = 0, which is the equation w + w n = uv ofthe three-fold W n on identifying i w = y + z n , w = 2 − /n z , as before. Thus U (1) rotates u with charge 1 and v with charge −
1, and we may lift this to an action on C with coordinates ( A , A , B , B , z ) by assigning charges (1 , , − , , x, y, u, v, ξ, µ ) under U (1) are (0 , , , − , − , u = v = ξ = 0 and u = v = µ = 0 – recall that ξ = 1 /µ on the overlap. Thuson the exceptional CP we fix the north pole ξ = 0, and also the south pole µ = 0. Wethus see that after resolving W n to W + n the fixed point set under U (1) is two disjointcopies of C , over the two poles of the CP . Indeed, recall that x = − vµ − Z n on thepatch H (where Z = z ), and thus the fixed locus at v = µ = 0 is described by theequation x = − z n . Changing variables as above, this becomes precisely w = − i w n .Conversely, the fixed locus u = ξ = 0 is the equation y = − z n , which under the abovechange of variable becomes precisely w = i w n .One can also interpret this in the moment map picture. The moment map is µ = | A | − | B | . Turning on ζ , we also have (2.12). The exceptional CP is, for ζ >
0, at B = B = 0. Then the moment map restricted to CP becomes simply µ | CP = | A | .But also | A | = ζ − | A | on this locus, and thus we see that on CP the moment mapranges from µ = 0 at A = 0 to µ = ζ at A = 0. These are precisely the two polesof the CP , which is where the fixed locus is. We thus see that the CP is mappedto an interval in the image of the moment map µ , which recall is the R direction,with the endpoints of the interval being where the NS5-branes are after performing theT-duality along U (1) . Notice that in the holomorphic picture A = 0 is the south pole µ = 0 while A = 0 is the north pole ξ = 0. For negative parameter ζ <
0, the rolesof A i and B i swap. In this case we will have coordinates ˜ ξ = B /B and ˜ µ = B /B on the exceptional CP , which is now located at A = A = 0. The moment mapis µ | f CP = −| B | . This ranges from 0 at B = 0 to − ζ at B = 0, with the twoendpoints being the NS5-brane loci. Notice that the brane at − ζ is B = 0, which is˜ ξ = 0, which is the same NS5-brane that moves for ζ >
0, namely that with w = i w n .To conclude, we see that the T-dual of resolving W n to W ζn is simply to separate thetwo NS5-branes in the R direction by a distance ζ – they are wrapped on the same29urves as before in the C direction. In terms of Figure 3, the NS5-brane on theleft hand side moves a distance ζ in the (transverse, as drawn) R direction. Noticethat once we resolve W n there is no canonical place to put the D3-branes – we haveto pick a point on W ζn . It is natural (in the sense that it preserves a U (1) ⊂ SU (2) r symmetry) to put them either at the north pole or south pole of the CP , in whichcase the D3-branes intersect either one NS5-brane or the other.Figure 4: On the left hand side: the positions of the two NS5-branes with resolutionparameter ζ in the Type IIA dual. The NS5-brane at position ζ is that wrapped on w = i w n , while the brane at the origin is that wrapped on w = − i w n . On the righthand side: the positions of the 5-branes after turning on the RR flux in the Type IIAdual, which fibres the resolution parameter over the R direction. One of the branesrotates so that they now intersect at the origin of the R − R plane.We may now consider what happens when we turn on the RR two-form flux. Recallthis fibres the parameter ζ over the R direction in Type IIA. It is simple to see whatthis does in the IIB brane picture. Consider a fixed point in R , which means fixing aparticular value for ζ . Then the 5-branes are separated by some distance ζ in the R direction. More precisely, the above analysis shows that for ζ > R , while the brane at the north pole is at ζ in R . As we move towards the origin in R , the 5-branes get closer together in the R direction, until finally at the origin they meet. We may then pass through the originto ζ <
0, where the behaviour is the same (with A i replaced by B i ). This shows thatafter turning on the RR two-form flux, the 5-branes rotate from being at fixed paralleldistance in the R direction (and filling the R direction), to being two lines in the R − R plane that cross at the origin – see Figure 4. This means that, after turningon the RR two-form flux, the 5-branes meet precisely at the origin in R . although30hey are still non-trivially holomorphically embedded in C as w = ± i w n .Notice that for n = 1 the above indeed reproduces the Type IIB brane picture inABJM [4] – up to two important details. First, in the case n = 1 we have derived theType IIB brane dual by starting with C / Z k , reducing to Type IIA along U (1) b and thenT-dualizing to Type IIB along U (1) . In [4], the authors instead began with the TypeIIB brane picture, and argued that T-dualizing to Type IIA and uplifting to M-theorygave a non-trivial hyperk¨ahler eight-manifold as the uplift, which is characterized bytwo harmonic functions, defined on two copies of R . The difference between thesetwo pictures is that the former is simply the near-brane limit of the latter. Indeed,ABJM showed explicitly that the near-horizon limit of the hyperk¨ahler manifold indeedgives C / Z k , which amounts to dropping the non-zero constant term in the harmonicfunctions. This is the dual geometry in the region near to where the 5-branes intersectat the origin in R (which are the two copies of R mentioned above).Second, and more importantly, in the ABJM brane picture the rotated 5-brane inFigure 4 is in fact a bound state of an NS5-brane with k D5-branes – the latter iseffectively the T-dual of the k units (3.47) of RR two-form flux through the (fibred)exceptional CP in the Type IIA geometry. To see the presence of the k D5-branesin the (1 , k )5-brane bound state directly is not straightforward in the discussion wehave given above. However, the k units of D5-brane charge can be seen indirectly byconsidering a certain tadpole. Thus, we begin in Type IIA on C ( M ), which recall mayalso be thought of as W ζn fibred over R . Pick a non-zero point in R , and considerthe exceptional CP of size | ζ | in W ζn over this point. If we wrap a D2-brane over this CP , we get a point particle in R , . However, because of the k units of RR two-formflux (3.47) through this CP , in fact this configuration does not exist in isolation: onemust have k fundamental strings ending on the wrapped D2-brane. To see this, notethe Wess-Zumino coupling on the D2-brane:1(2 π ) l s Z R time A Z CP F = k · g s πl s Z R time A . (4.49)To cancel this tadpole, we precisely require k fundamental strings to end at a point onthe CP .Consider the T-dual to this in Type IIB. As already discussed, the exceptional CP maps to an interval in the R direction, between the two 5-branes: this lies at thechosen point in R , and is at the origin in C . A D2-brane wrapped on the CP thusT-dualizes to a D1-brane stretched between the two 5-branes in the R direction. The k fundamental strings ending on the D2-brane T-dualize to k fundamental strings ending31igure 5: On the left hand side: the naive T-dual configuration to a D2-brane wrappedon the CP at a fixed non-zero point in R is a D1-brane stretching between thetwo NS5-branes, with k fundamental strings also ending on the D1-brane and one ofthe NS5-branes to cancel the tadpole. On the right hand side: the correct T-dualconfiguration, in which the D1-brane and k fundamental strings form a (1 , k ) stringbound state, which then must necessarily end on a (1 , k )5-brane. (Notice that theD1-brane must also wind around the S circle as one moves from one 5-brane to theother along its worldvolume.)on the D1-brane. In particular, the fundamental strings may end at one of the polesof the CP . In the IIB picture, we therefore have a D1-brane and also k fundamentalstrings terminating on one of the 5-branes (while for the other 5-brane there is onlya D1-brane ending on it). In general, a ( p, q ) string, where p denotes the number ofD1-branes and q the number of fundamental strings in a bound state string, can onlyend on a ( p, q )5-brane. Thus the only way to make sense of the above tadpole is thatthe 5-brane is in fact a (1 , k )5-brane, and the D1-brane and k fundamental strings forma (1 , k ) bound state ending on this. Of course, this precisely reproduces the correctbrane configuration of ABJM in the case of n = 1.To conclude, we have shown that M-theory on X n / Z k has a Type IIB dual of Hanany-Witten type: it is identical to the brane set-up for n = 1 described by ABJM [4], exceptthat the 5-branes are wrapped on the holomorphic curves w = ± i w n inside C –see Figure 6. Having described the Type IIB brane dual, an important dynamical question is whathappens when we move the two 5-branes past each other on the S circle. This was32igure 6: The final Type IIB dual of M-theory on X n / Z k . The spacetime is R , × R × S × R × C . There are N D3-branes filling the R , directions and wrappingthe S circle; they are at the origin in R , R and C . There are also two spacefilling5-branes in R , at points on the S circle. The first is an NS5-brane, sitting at theorigin in R and filling R , which wraps the curve w = − i w n in C . The second isa (1 , k )5-brane, wrapping an angled line through the origin in the R − R plane, andwrapping the curve w = i w n in C .first studied by Hanany-Witten [14], and the analysis in section 5 of that paper maybe applied directly to the case n = 1 (the ABJM case). We thus begin by describingthe n = 1 case, and then explain how to apply this result for n > C so that the brane intersections in R are normal crossings.We thus start with n = 1. We suppress the spacetime R , from the discussion, sinceall branes are spacefilling in these directions. Thus the relevant geometry is S × R .We have an NS5-brane at a point 0 = t ∈ S and at the origin in 789, and a (1 , k )5-brane at the origin 0 ∈ S and at the origin in 345. Notice that we have, for convenienceof notation, rotated the axes relative to Figure 6: the argument we are about to giveis entirely topological, and so is unaffected. We denote these submanifolds as W NS,t and W (1 ,k ) , respectively. These two copies of R that are wrapped by the 5-branes thusintersect normally at the origin in R . However, importantly, the branes do notactually intersect in spacetime unless t = 0.The (1 , k )5-brane sources k units of RR three-form flux F through a sphere S linking its worldvolume. Thus, let S be a normal sphere around a point on the (1 , k )-brane in S × R , so that 1(2 πl s ) g s Z S F = k . (4.50)33ollowing [14], we then define the linking number L t = 1(2 πl s ) g s Z W NS,t F . (4.51)This is independent of t as t is varied, provided we do not cross the origin t = 0. Thereason for this is that F is closed on the complement of the (1 , k )5-brane worldvolume,and the independence of (4.51) on t then follows from Stokes’ Theorem. More precisely,d F is a four-form which is supported only on the (1 , k )5-brane worldvolume at t = 0and the origin in 345: it is k times a delta-function representative of the Poincar´e dualof W (1 ,k ) .Consider now moving the NS5-brane from t + >
0, on the right of the (1 , k )5-brane,to t − < I = [ t − , t + ] be the interval in the S circle covered in thismotion. Then we have linking numbers (4.51) L + and L − on the right and left. Wemay compute the change in linking number using Stokes’ Theorem: L + − L − = 1(2 πl s ) g s Z W NS × I d F = k . (4.52)On the worldvolume of the NS5-brane there is a U (1) gauge field A NS , with fieldstrength F NS , and it is only the combination Λ = C − πl s F NS that is gauge invariant.Moreover, F | W NS = dΛ , (4.53)meaning that F must be exact on the NS5-brane worldvolume W NS,t . In the non-compact setting of interest, of course all closed forms are exact on W NS,t ∼ = R , so(4.53) is always satisfied. However, what we learn from (4.52) is that the period of F through W NS,t changes by k units as we move the NS5-brane from the right t > t < , k )5-brane. The explanation for this is that k spacefilling D3-branesare created at the intersection point t = 0 when the branes are moved past each other.Indeed, such a D3-brane ending on the NS5-brane is a delta-function source for F NS :12 πg s d F NS = ± δ ( p ) (4.54)where p ∈ W NS ∼ = R . That is, the D3-brane ending on the NS5-brane is a magneticmonopole for this U (1) gauge field. The sign in (4.54) depends on whether the D3-braneends from the right or from the left on the S circle, which it wraps (a monopole oranti-monopole). Integrating k times (4.54) over W NS precisely accounts for the changein linking number (4.52). This is the Hanany-Witten effect.34aving carefully reviewed this effect, we may now apply it to the case with n > n > C : theycross at a single point at the origin, but they are wrapped on non-trivial curves. Wemay remedy this by deforming the curves that the 5-branes are wrapped on. Thus, wechange w = − i w n −→ w = − i n Y i =1 ( w − α a ) + α (4.55) w = i w n −→ w = i n Y i =1 ( w − β a ) + β . (4.56)Here α a , β a , a = 0 , . . . , n , are arbitrary parameters. The point of these deformationsis that ( a ) they preserve the boundary conditions at infinity, since we have addedonly lower order terms to the polynomials, and ( b ) the resulting curves now intersectnormally in C . Indeed, these two curves in C intersect where the w coordinatein (4.55) equals the w coordinate in (4.56). This results in the n th order polynomiali n Y i =1 ( w − α a ) + i n Y i =1 ( w − β a ) − α + β = 0 . (4.57)For generic values of the parameters α a , β b , this will have precisely n solutions for w , say w ( i )0 , i = 1 , . . . , n . Thus the resulting curves generically intersect at n points( w ( i )0 , w ( i )1 ), where of course w ( i )1 is given by (4.55) (or (4.56)) evaluated at w ( i )0 . More-over, the intersects of the curves near to these n points look precisely like the linear n = 1 case.We are now in good shape: after this generic deformation that preserves the bound-ary conditions of the branes at infinity, the two branes intersect ordinarily at n pointsin R (they always cross at the origin of the R − R plane). The above discussion ofthe Hanany-Witten effect shows that the creation of the k D3-branes as an NS5-branecrosses a (1 , k )5-brane occurs entirely locally at the points where the branes intersect inspacetime. Thus if we move our deformed NS5-brane past the deformed (1 , k )5-brane,we obtain precisely n copies of the n = 1 result, i.e. in total nk D3-branes are createdas they are moved past each other. More precisely, k D3-branes are created at each ofthe n points ( w ( i )0 , w ( i )1 ) (at the origin in the R − R plane, and stretched along the S circle). Notice that this result is independent of the choice of deformation parameters α a , β a , as it is topological. Thus after moving the branes past each other we maydeform back to α a = β a = 0, where the nk created D3-branes are all at the origin in R . 35 .4 The field theory duality The brane creation effect described in the last section leads to an interesting fieldtheory duality, discussed for the ABJM theory in [23], [32]. Here we briefly describethe situation for general n . We begin with the Type IIB brane set-up corresponding tothe gauge group U ( N + l ) k × U ( N ) − k . This is shown on the left hand side of Figure 7.Figure 7: On the left hand side: the initial brane configuration, with ( N + l ) D3-branessuspended between the 5-branes on one side of the S circle, and N D3-branes onthe other. On the right hand side: moving the NS5-brane anti-clockwise around thecircle pulls the l fractional branes with it. After passing the (1 , k )5-brane these swaporientation, becoming l anti-branes, and in addition nk D3-branes are created.Consider, without loss of generality, moving the NS5-brane around the circle. Ro-tating it anti-clockwise by one revolution, as shown on the right hand side of Figure 7,the gauge groups become U ( N ) k × U ( N + nk − l ) − k . In particular, we note that the U ( N + nk ) k × U ( N ) − k theory can be deformed to the U ( N ) k × U ( N ) − k theory in thisway, which is the required field theory duality to match the dual supergravity analysismentioned at the very end of section 2.2. Moving the NS5-brane multiple times aroundthe circle, or in the other direction, apparently leads to further equivalences, as ob-served for the n = 1 ABJM theory in [23]. This certainly deserves further careful studyof the brane system to understand properly, although we shall make some commentson this in section 6.2. 36 The deformed supergravity solution
In this section we describe a supergravity solution [15] which is a deformation of theAdS × V , / Z k M-theory background discussed in section 3.1, in the sense that itapproaches the latter asymptotically at infinity. Throughout this section we set n = 2.We also begin with k = 1, and restore general k later. T ∗ S We begin by describing a deformation of the Calabi-Yau cone metric on the quadriccone X . The latter has an isolated singularity at z = · · · = z = 0 that may be deformed to a smooth non-compact Calabi-Yau variety X , diffeomorphic to T ∗ S (the cotangent bundle of S ), via X ≡ ( X i =0 z i = γ ) , (5.58)where γ ∈ C is a constant. For γ = 0 this describes a smooth complex structureon T ∗ S . The deformation breaks the C ∗ ∼ = R + × U (1) R symmetry of the cone to Z ⊂ U (1) R . Using the broken U (1) R action we take γ ∈ R + in what follows. The S = SO (5) /SO (4) zero-section is then realized as the real locus of X in C . Thecotangent bundle structure may be seen explicitly by writing z i = cosh (cid:0) √ p j p j (cid:1) x i + i √ p j p j sinh (cid:0) √ p j p j (cid:1) p i . (5.59)Then P i =0 x i = γ , P i =0 x i p i = 0, so that the S is { p i = 0 } .There is an explicit complete Ricci-flat K¨ahler metric on X which is asymptotic tothe cone metric at large radius, called the Stenzel metric. This is cohomogeneity oneunder the action of SO (5), with principal orbits diffeomorphic to V , = SO (5) /SO (3),and degenerate special orbit S = SO (5) /SO (4). The K¨ahler structure induces thestandard symplectic structure on T ∗ S , and thus the S is Lagrangian; in fact it isspecial Lagrangian, and is thus a minimal volume representative of the generator of H ( X , Z ) ∼ = Z . Note that given any Ricci-flat metric d s , the rescaled metric γ d s isalso Ricci-flat, for any positive constant γ ∈ R + , and this is essentially the constant γ above, which is proportional to the radius of the S . In the same sense as the more familiar deformed conifold in six dimensions.
37n terms of invariant one-forms on the coset space V , = SO (5) /SO (3), the metricon X may be written asd s X = c d r + c ν + a X i =1 σ i + b X i =1 ˜ σ i , (5.60)where a = 13 (2 + cosh 2 r ) / cosh r , b = 13 (2 + cosh 2 r ) / sinh r tanh r ,c = (2 + cosh 2 r ) − / cosh r . (5.61)More details may be found in appendix B. In these coordinates, the S is located at r = 0. Note here we have picked a particular representative metric in the conformalclass of metrics on X , i.e. a particular value of γ . It will be straightforward toreintroduce this scale later. The calibrated S in the above solution has fixed size,with induced round metric d s S = 3 − / ( ν + X i =1 σ i ) . (5.62)After a change of variable ρ ∼
169 12 / e r , (5.63)the asymptotic form of the metric isd s ≈ d ρ + ρ " X i =1 (cid:0) σ i + ˜ σ i (cid:1) + 916 ν + 2 / ρ / X i =1 (cid:0) σ i − ˜ σ i (cid:1) + . . . . (5.64)The leading term is the metric on the cone over the manifold Y = V , .For later use we record here the results of certain integrals. Noticing that the S isparametrized by ν, σ i , and recalling that V , is an S bundle over S , we have Z S ˜ σ ∧ ˜ σ ∧ ˜ σ = 2 π . (5.65)This is the volume of a unit S , as necessarily follows since the collapse of this S atthe S zero-section is regular. Writing the volume form of V , asdvol V , = 3 σ ∧ σ ∧ σ ∧ ˜ σ ∧ ˜ σ ∧ ˜ σ ∧ ν , (5.66)and using the total volume of V , (3.22), we deduce also that Z S ν ∧ σ ∧ σ ∧ σ = 8 π , (5.67)which is in fact the volume of a unit radius round S .38 .2 The deformed M2-brane solution The AdS × V , supergravity solution admits a smooth supersymmetric deformation,based on the above Stenzel metric. This solution was presented in [15]. We have foundand corrected a few minor mistakes in the formulas in [15], which are important forthe physical interpretation. The d = 11 solution is d s = H − / d s R , + H / γ d s X ,G = d x ∧ d H − + mα , (5.68)where m is a constant, d s X denotes the Stenzel metric, and α is a harmonic self-dual four-form on X [15]. In terms of the orthonormal frame (B.112) defined in appendixB this reads α = 3cosh r (cid:16) e ˜0123 + e (cid:17) + 12 1cosh r ǫ ijk (cid:16) e ij ˜ k + e ˜0 i ˜ j ˜ k (cid:17) . (5.69)More precisely, this is an L -normalizable primitive harmonic (2 , X . Notethat α generates H ( X , R ) ∼ = R . By the general results of [40], this is the only L -normalizable harmonic form on X in fact. The equation of motion for the G -fieldd ∗ G = 12 G ∧ G , (5.70)implies the following equation for the warp factor∆ X H = − m cosh r . (5.71)Here ∆ X denotes the scalar Laplacian on the Stenzel manifold with metric d s X . Thiscan be integrated explicitly in terms of the variable y = 2 + cosh 2 r , giving H ( y ) = − m √ Z d y ( y − / , (5.72)where an integration constant has been fixed by requiring regularity near to r = 0. Interms of the variable ρ introduced in (5.63), the asymptotic expansion reads H ( ρ ) = 2 m ρ + . . . for ρ → ∞ . (5.73)Notice that this has a different behaviour from the Klebanov-Strassler solution, whereone has logarithmic corrections. As explained in [15], this difference comes from thefact that the self-dual harmonic form is normalizable here, while it is not normalizablein six dimensions. At large ρ the solution becomes of the form (3.23), where here theAdS radius is expressed in terms of the integration constant m as R = m . We have introduced an explicit deformation parameter γ which is set to unity in [15]. Thismeasures the radius of the S at the origin. .3 The G -flux We now wish to discuss the quantization of the flux, thus relating the constant m to the quantized fluxes. Because the background is asymptotically AdS × V , , it isnatural to quantize the flux of ∗ G through the V , at infinity, as in (3.24), and interpretthis as the number of M2-branes in the UV. More generally, we may define a “running”number of M2-branes N ( r ) as N ( r ) = 1(2 πl p ) Z Y r ∗ G , (5.74)where the integral is evaluated on a seven-dimensional surface of constant r , which isa copy of V , . To compute this, we may use the four-form equation of motion (5.70)to write Z Y r ∗ G = 12 Z X r G ∧ G = 12 Z X r m | α | dvol X , (5.75)where the integral is evaluated on the Calabi-Yau X cut off at a distance r . The resultis N ( r ) = 1(2 πl p ) m vol( V , ) tanh r . (5.76)We see that this running number of M2-branes becomes a constant at infinity, where N ≡ N ( ∞ ) = 1(2 πl p ) m vol( V , ) . (5.77)This determines m in terms of the physical paramater N . Eliminating m we see thatthe (UV) AdS radius takes exactly the form (3.25).We are not quite done, however. There is a non-trivial cycle in the geometry, namelythe four-sphere at the zero-section of X = T ∗ S . Thus we have to impose the quanti-zation of the four-form flux through this cycle. Noting that the restriction of the (2 , α to a four-sphere at any distance r from the origin is α | S r = 1 √ r ν ∧ σ ∧ σ ∧ σ , (5.78)we compute 1(2 πl p ) Z S G = 1(2 πl p ) m √ π M ∈ N , (5.79)40here recall that the volume of the unit S at the origin is 8 π /
3. The reason fordenoting the integer flux as ˜ M will become clear momentarily. We hence obtainanother expression for m , namely m = 27 π l p ˜ M . The running number of M2-branesthen takes the simple form N ( r ) = ˜ M r . (5.80)There is a simple way to check the numerical factor here. If we integrate (5.70) overthe whole of X , the left hand side gives (2 πl p ) N . On the other hand, the right handside is a topological quantity. To see this, note that the integral of G over S is bydefinition (2 πl p ) ˜ M . But we may also regard G as defining an element of H ( X , R ).The map R ∼ = H ( X , R ) → H ( X , R ) ∼ = R is just multiplication by 2, the latterbeing the Euler number of S . Then we may interpret R X G ∧ G as the cup product H ( X , R ) × H ( X , R ) → H ( X , R ) = R via [ G ] ∪ [ G ] cpt = (2 πl p ) ˜ M · ˜ M . This isa simple topological check on (5.80).Since we have N = ˜ M /
4, and N must be an integer, we have to set ˜ M = 2 M . Wethus obtain the relation N = M , (5.81)where 2 M is the number of units of G -flux through the S (5.79). Notice that the higherderivative X term in M-theory would lead to a O (1 /N ) correction to this formula. Infact an explicit solution, generalizing that above and including the X correction, wasgiven in [41] . Of course, the supergravity solution is only valid at large N (and hencelarge M ) in any case, and this term is a subleading correction.As a consequence of the relation ˜ M = 2 M we also see that there is no torsion G -fluxturned on in H ( V , , Z ) ∼ = Z . To see this we recall that there is a relation between thecohomology of the deformed space X and the cohomology of its boundary ∂ X = V , .The only non-trivial cohomology of X is H ( X , Z ) ∼ = H ( X , Z ) ∼ = Z , the latter beinggenerated by the S zero-section. There is a map Z ∼ = H ( X , Z ) → H ( V , , Z ) ∼ = Z induced by restriction to V , = ∂ X which is simply reduction modulo 2. Thecalculation (5.79) means that as a cohomology class [ G ] = 2 M e , where e denotes thegenerator of H ( X , Z ). This then maps [ G ] → ∈ H ( V , , Z ) ∼ = H ( V , , Z ) ∼ = Z . It is again important here that the membrane anomaly on X vanishes. This follows from the factthat w ( X ) | S is twice the fourth Stiefel-Whitney class of the bundle T S , and hence zero mod 2(the latter Stiefel-Whitney class also happens to be zero). Although some errors in [15] have propagated to this reference.
41e may also define a “running C -field period”. Recall that V , may be thought ofas an S bundle over S . Then the generator of H ( V , , Z ) ∼ = H ( V , , Z ) ∼ = Z may betaken to be a copy of the S fibre at a fixed point on the base S . We can identify thetorsion three-cycle at a distance r as the three-sphere at a distance r from the originof the fibre R , at a fixed point on S . We have α | R = sinh r √ r d r ∧ ˜ σ ∧ ˜ σ ∧ ˜ σ , (5.82)and thus c ( r ) ≡ πl p ) Z S r C = m (2 πl p ) Z R r α = M (cid:20) r (cid:18) r − (cid:19) + 2 (cid:21) . (5.83)Notice that c ( ∞ ) = M . Indeed, this is again purely a topological integral, namely(1 / (2 πl p ) ) R R G = M , and shows that the holonomy of the C -field on V , at infinityis indeed trivial, cf (3.26). Z k quotient If we wish to consider deformations of the V , / Z k supergravity background with k > X / Z k is then singular, having two isolated C / Z k singularitiesat the north p N and south p S poles of the S zero-section. Since we cannot trust thesupergravity solution near to these points, we should remove them from the spacetimein any supergravity analaysis. It then makes sense to analyse flux quantization onthe smooth manifold ( X \ { p N , p S } ) / Z k . This has a boundary with three connectedcomponents: V , / Z k at infinity, and two copies of S / Z k near to r = 0.Since H ( X , Z ) ∼ = Z , generated by the S zero-section, it follows from a simpleMayer-Vietoris sequence that also H ( X \ { p N , p S } , Z ) ∼ = Z . On removing the twopoints, the image of the S zero-section in X \ { p N , p S } is I × S , where I is an interval.Thus the image of this S naturally gives a relative class in H ( X \{ p N , p S } , S ∐ S , Z ),although again it is simple to show that this is isomorphic to H ( X \ { p N , p S } , Z ) andthus the relative class is represented by a closed 4-cycle also.Consider a Z k -invariant closed four-form G on X that has non-zero integral over the S . Then one obtains a four-form on ( X \ { p N , p S } ) / Z k with non-zero integral over I × S / Z k , where Z k acts along the Hopf fibre of the S . We now normalize the flux G/ (2 πl p ) to have period ˜ M ∈ Z through this (relative) 4-cycle. It follows that liftingto the covering space X , we obtain a period k ˜ M through S . Then the integral of422 πl p ) − G ∧ G over the covering spacetime X may be carried out as in the smoothcase, to give · ( k ˜ M ) · ( k ˜ M ) = k M . Thus on the quotient X / Z k we obtain N = 1(2 πl p ) Z V , / Z k ∗ G = 1(2 πl p ) Z X / Z k G ∧ G = kM . (5.84)Similarly, we have 1(2 πl p ) Z R / Z k G = 1(2 πl p ) Z Σ C = M , (5.85)where we have noted that the generator Σ of H ( V , / Z k , Z ) ∼ = Z k is given by a copy ofthe boundary of the R / Z k fibre of T ∗ S / Z k over the north pole p N ∈ S . Comparingto (3.26), we see that l ∼ = 0 mod 2 k at infinity, and hence there are no fractionalM5-branes . Clearly, this is in stark contrast to the Klebanov-Strassler solution. The deformed supergravity background that we have discussed is of a type which has noknown counterpart in the context of the AdS /CFT correspondence. This was alreadynoticed in [15, 19, 42]. The UV region is asymptotic to a Freund-Rubin backgroundAdS × Y , and thus according to the AdS/CFT dictionary it should be dual to theconformal Chern-Simons-quiver theory extensively discussed in the paper. On theother hand, in the IR region the solution is smooth and displays a finite-sized minimalsubmanifold at the bottom of the throat. Therefore, according to the general rules ofgauge/gravity duality, the dual field theory should have a mass gap and is presumablyconfining [43]. Understanding the precise mechanism in the field theory is clearly aninteresting challenge. In this final section we take a few steps in this direction, leavinga more detailed investigation for future work. As we have already explained, at infinity the deformed solution approaches the AdS × V , / Z k background. Since H ( V , / Z k , Z ) ∼ = Z k , at infinity we can only have a flattorsion G -flux of [ G ] = l mod 2 k . A careful examination of flux quantization in thedeformed solution leads to 2 M units of G -flux through the minimal four-cycle S / Z k at the zero-section r = 0. However, this G -field descreases as we move towards theUV, eventually disappearing at infinity r = ∞ . The topological class of this G -flux at43nfinity is [ G ] = 0, while the flux of ∗ G through V , is N = kM . This leads us toconjecture that the field theory in the UV is the superconformal Chern-Simons-quivertheory with gauge group U ( kM ) k × U ( kM ) − k . (6.86)Note that the ranks of the gauge groups could receive subleading corrections that maybe important for a consistent interpretation.On general grounds, the field theoretic interpretation of the deformation is either aperturbation by a relevant operator in the Lagrangian, or involves spontaneous sym-metry breaking. These two possibilities are distinguished by the asymptotic behaviourof perturbations in AdS . In order to use the AdS/CFT dictionary we need to writethe AdS metric in Fefferman-Graham coordinatesd s (AdS ) FG = 1 z (cid:0) d z + d x µ d x µ (cid:1) , (6.87)by changing coordinates ρ = 1 /z . Here recall that ρ is related asymptotically to r viathe change of variable (5.63). In particular, for scalar modes we then have ϕ ∼ ˆ ϕz − ∆ + ϕ z − ∆ , (6.88)with ϕ corresponding to perturbing by an operator of dimension ∆, and ˆ ϕ correspond-ing to the VEV of such an operator. Aided by the map between chiral multiplets inthe SCFT and modes in the Kaluza-Klein spectrum on V , , discussed earlier, we willsee that the former possibility is realized.To see this, we examine the leading behaviour of the G -field at infinity, and thecorresponding pseudoscalar mode in AdS . We may discuss this in the context ofgeneral Sasaki-Einstein solutions and then specialize to the case of interest. Consider aself-dual harmonic G -flux in the Calabi-Yau cone background R , × C ( Y ), of the form G = α = d( ρ − ν β ) , (6.89)where ρ is the radial variable on the cone. This implies ∆ Y β = ν β , where ∆ Y is theLaplace operator on Y acting on three-forms. For the associated AdS × Y solution,we may then consider a fluctuation of the type δC = π · β . It was shown in [44] thatthis leads to a pseudoscalar field π in AdS with mass m = ν ( ν − . (6.90) The reader should not confuse the mass m here with the paramter m in the deformed solution. −
3) = m , we obtain ∆ ± = (3 ± | − ν | ). Which branch to pick depends a priori on thespecific operator we consider. Going back to our particular G = α given by (5.69), wesee that β ∝ (cid:18) σ ∧ ˜ σ ∧ ˜ σ + 12 ǫ ijk σ i ∧ σ j ∧ ˜ σ k (cid:19) , (6.91)and ν = 4 /
3. Then ∆ + = 3 − ν = , while ∆ − = ν = . Now, going throughall the pseudoscalar modes undergoing shortening conditions in the tables in [11], wefind a mode with ∆ = while the other possibility is not realized. In particular, thismode arises as the pseudoscalar component of the chiral operators with dimensions∆ = m + 1, with m = 2, that we discussed in section 3.3. From the asymptoticscaling α ∼ z / , we conclude that this operator is in fact added to the Lagrangian (seealso [19]).Since this is the pseudoscalar component of a chiral superfield, we see that it is aFermionic mass term ψ α ψ α . This breaks parity invariance, which is reflected in thegravity solution in the presence of the internal flux, the latter being odd under parity.In general, such mass terms may be added to the Lagrangian, in a supersymmetricway, by a quadratic superpotential deformation δW = µ Tr[ φ ] ⇒ δ L = − ∂ δW∂φ i ∂φ j ψ αi ψ j α + . . . . (6.92) A priori , we have three such possible mass terms, compatible with the SU (2) r globalsymmetry of the deformed background, namely δW = µ + (cid:0) Tr[Φ ] + Tr[Φ ] (cid:1) + µ − (cid:0) Tr[Φ ] − Tr[Φ ] (cid:1) + µ Tr[ A B + A B ] . (6.93)where in the above we mean superfields.We may deduce which terms are present by analysing more carefully the symmetriesof the deformed solution. Recall from section 2.1 that in the undeformed field theorywe have a Z flip2 symmetry that exchanges Φ ↔ Φ , A i ↔ B i . The generator actson the z i coordinates, introduced just below equation (2.9), as ( z , z , z , z , z ) → ( − z , z , − z , z , − z ). Hence Z flip2 ⊂ O (5) acts on the deformed quadric (5.58). The This deformation then introduces various additional terms in the Lagrangian. For example, wehave a quadratic term µ Tr[ φ † φ ] in the bosonic F-term potential, with dimension ∆ = 4 /
3, as well aslinear terms in µ . Presumably these operators may be detected by analysing appropriate linearizedperturbations of the background. However, their structure should be constrained by supersymmetry.See [45] for discussion of a related issue in the context of mass deformations of the ABJM theory. G -flux then breaks this Z flip2 symmetry. To see this, notice that for k = 1 thezero-section of X = T ∗ S is S , embedded in R by the real parts of the z i coordinatesin (5.58). The volume form on S may be writtenvol( S ) = 14! ǫ ijklm z i d z j ∧ d z k ∧ d z l ∧ d z m | { P i =0 z i = γ , z i ∈ R } . (6.94)This hence changes sign under the generator of Z flip2 . Now since Z flip2 is an isometry, itnecessarily maps L harmonic forms to L harmonic forms, and as mentioned earlierthe results of [40] imply that G int ∝ α (5.68), where α is given by (5.69), is the onlysuch form. Thus the generator of Z flip2 maps α
7→ ± α . But since α restricts to thevolume form on S at r = 0, we see that the generator of Z flip2 maps α
7→ − α , and thus G int
7→ − G int . Hence the related superpotential deformation in (6.93) should also beodd. This requires that µ + = µ = 0, leaving precisely the following supersymmetricmass-term W → W + µ (cid:0) Tr[Φ ] − Tr[Φ ] (cid:1) . (6.95)We may then regard the full superpotential as depending on the two parameters s and µ . Notice that by setting s = 0, the mass term µ is precisely that leading to the ABJMtheory in the IR, after integrating out the adjoints.The deformed F-term equations following from the superpotential deformation (6.95)read B i Φ + Φ B i = 0 , (6.96)Φ A i + A i Φ = 0 , (6.97)3 s Φ + ( B A + B A ) + µ Φ = 0 , (6.98)3 s Φ + ( A B + A B ) − µ Φ = 0 . (6.99)The simple linear change of variableΦ = Ψ − µ s , Φ = Ψ + µ s (6.100)then leads to B i Ψ + Ψ B i = 0 , (6.101)Ψ A i + A i Ψ = 0 , (6.102)3 s Ψ + ( B A + B A ) = µ s , (6.103)3 s Ψ + ( A B + A B ) = µ s . (6.104)46n particular, we see that the Abelian moduli space is exactly the deformed singularity(5.58). The deformation parameter is proportional to the mass, γ = µ / s . The supergravity solution implies that the N = 2 superconformal Chern-Simons-matter theory deformed by the mass term will flow in the IR to a confining theory. Weleave a field-theoretic understanding of this for future work, restricting ourselves hereto making only some preliminary comments in this direction.Firstly, it is instructive to contrast the pattern of U (1) R symmetry breaking of oursolution with that of the Klebanov-Strassler theory. In the latter case the U (1) R sym-metry is broken to Z M in the UV by the chiral anomaly, and this is then spontaneouslybroken to Z , yielding M vacua. On the gravity side, the breaking of U (1) R to Z M isreflected by the non-invariance of the fluxes already in the UV [18, 46]. The M vacuaare then reflected by the presence of supersymmetric probe branes, representing BPSdomain walls interpolating between the vacua. In three dimensions there is no chiralanomaly, and thus U (1) R cannot be broken in this way. Indeed, in the supergravitysolution we discussed the parameter M is not a UV parameter that one can dial atinfinity, and in fact the flux vanishes asymptotically. We also expect that no wrappedbranes will give rise to BPS domain walls, although we have not checked this.In analogy with the Klebanov-Strassler cascade, one possible way to interpret theRG flow described by the supergravity solution is to imagine that once the conformaltheory is deformed by the mass term in the UV, it starts “cascading”, going througha sequence of Seiberg-like dualities where the ranks of the gauge groups decrease,until in the deep IR perhaps one gauge group disappears, and the low energy-theoryconfines. This idea has recently been suggested in [32, 47] in the context of ABJM-like theories, although the models studied in these references are different from ourmodels. This interpretation is motivated by the brane creation mechanism that wediscussed in section 4.4, and by the fact that in the solution there is a varying B -field (in the Type IIA reduction). More precisely, the B -field suggests that as weproceed to the IR, the NS5-branes rotate around the circle. Taking this point ofview, and applying the duality rule of section 4.4, we end up in the IR with a gaugegroup U ( − kM ) k × U ( kM ) − k after M steps, which clearly doesn’t make sense sinceone gauge group has negative rank. (We could of course stop applying the dualityat the previous step.) Notice, however, that what is the precise gauge group in the47R depends on the starting point in the UV, which in turn depends on subleadingcorrections to kM . In any case, it is not clear whether applying this rule is correct,once we turn on the mass deformation. In fact, more conservatively, given the massterm one should integrate out the heavy degrees of freedom, and obtain an effectivelow-energy theory in the IR. In principle this theory should then exhibit confinement(without supersymmetry breaking). Integrating out the Fermions would a priori leadto a possible shift of the Chern-Simons levels. However, because the Fermions are inthe adjoint representation in fact the levels are not shifted. Indeed, we have alreadynoted that the mass term is exactly the same mass term which produces the ABJMtheory at low energy, starting from the Chern-Simons theory in Figure 1 with s = 0.Integrating out the bosonic components of the chiral fields in the mass-deformed n = 2theory, the effective superpotential for the low-energy fields A i , B i results in a non-local expression, involving square roots of polynomials in these fields. Hopefully, furtherwork along these lines will lead to a precise identification of the IR field theory. In this paper we have constructed a new example of AdS / CFT duality by proposinga simple N = 2 Chern-Simons-matter quiver field theory as the holographic dualto the AdS × V , / Z k Freund-Rubin background in M-theory. This duality presentsseveral novel aspects. For example, the geometry, and hence the field theory, has an SU (2) × U (1) × U (1) R global symmetry (enhanced to SO (5) × U (1) R for k = 1), andhence these models are non-toric. Examples of AdS/CFT dual pairs of non-toric type,where both sides are known explicitly, are quite rare. This model may be thought ofas describing the low-energy theory of multiple M2-branes at a quadric hypersurfacesingularity. In fact, this is the n = 2 member of a family of hypersurface singularities( A n − four-fold singularities), labelled by a positive integer n , for which we have alsopresented the corresponding field theories. However, we have explained that only for n = 2 and n = 1 do these singularities give rise to Freund-Rubin AdS duals, the n = 1model being the ABJM theory. We note that [12] discussed the larger class of ADEfour-fold singularities, and it was shown in this reference that in this class the onlycases that can admit Ricci-flat K¨ahler cone metrics are A = C , A and D . It wouldbe interesting to construct Chern-Simons-matter theories dual to other hypersurfacesingularities, and to see whether the D theory admits a Freund-Rubin holographicdual, analogous to that discussed in this paper.48n this paper we have considered the case where the Chern-Simons levels are equal k = − k = k . Relaxing this condition, thus allowing for arbitrary levels, correspondsto deforming the Type IIA solutions that we discussed in section 3.2 by turning on aRomans mass [20]. Such solutions will be similar to those discussed in [51, 52] and itwould be interesting to find these solutions explicitly.Another interesting aspect of the model we discussed is that there exists a deformedsupergravity solution, that we have argued corresponds to a particular supersymmetricmass deformation of the conformal theory. This deformation is similar to those studiedin [48, 49, 45] and other references. We have seen that this mass term is dual to aharmonic (2 , G -flux on the Calabi-Yau geometry. Quiterecently the authors of reference [50] have shown how self-dual background fluxes in-duce mass terms in the M2-brane worldvolume action, and it would be interesting tosee whether this construction generalizes to N = 2 backgrounds of the type we havestudied. In the present context the effect of this mass term is rather different fromthat in the ABJM model studied in [48, 49, 45]: it deforms the classical moduli spacein a way that precisely matches the geometry in the supergravity dual. In particular,the solution develops a finite-sized S in the IR, implying that the theory becomesconfining. Motivated by brane constructions, we have briefly discussed how this de-formation might be interpreted as a “cascade”, analogous to the Klebanov-Strasslercascade. However, further work is needed in order to obtain a more conclusive inter-pretation of the RG flow, and in particular a clearer understanding of the field theoryin the deep IR. We expect a similar story to repeat for other deformed solutions withself-dual G -flux, based on different special holonomy manifolds [42, 15].Finally, in appendix C we describe a Type IIA reduction of the supergravity solutionsthat is different to that considered in the main text, i.e. we reduce on a different choiceof M-theory circle. On general grounds, one expects this to lead to a field theory thatis mirror to that considered in section 2 (see, for example, [53]). It would be interestingto study this reduction further. Acknowledgments
We are very grateful to A. Armoni, S. Gukov, P. Kumar, J. Maldacena, C. Nu˜nez,M. Piai, Y. Tachikawa and D. Tong for discussions. J. F. S. is supported by a RoyalSociety University Research Fellowship. 49
Some cohomology computations
In the main text we have made use of a number of different cohomology groups of thevarious manifolds we have defined, and also the relations between the groups. In thisappendix we present the relevant computations.We begin by defining a manifold that does not appear in the main text: we define X n by X n = ( n Y γ =1 ( z − a γ ) + X i =1 z i = 0 ) ⊂ C . (A.105)Here the a γ , γ = 1 , . . . , n , are real, pairwise non-equal constants, which we order as a < a < · · · < a n . The manifold X = X in the main text, which is the deformationof the quadric singularity. The X n are smooth non-compact complex manifolds withboundaries ∂ X n = Y n , where Y n is defined by (2.10), (3.18). Indeed, the X n are deformations of the X n singularities (2.10).The cohomology of X n was discussed in [13], and we briefly review their analysis.For γ = 1 , . . . , n − S γ by requiring that z is realwith a γ < z < a γ +1 , and that the z i , for i = 1 , . . . ,
4, are all real or all imaginary,depending on the value of γ mod 2. These n − H ( X n , Z ) ∼ = Z n − ∼ = H ( X n , Y n , Z ), where the last step is Poincar´e-Lefschetz duality. This is theonly non-trivial homology group of X n (of course H ( X n , Z ) ∼ = Z ). Each four-sphere hasself-intersection number 2, since its normal bundle may easily be seen to be T ∗ S whichhas Euler number 2, and by construction the intersection number of S γ with S γ +1 is 1,with all other intersection numbers vanishing. Poincar´e-Lefschetz duality implies that H ( X n , Y n , Z ) and H ( X n , Z ) are dual lattices, where recall that f : H ( X n , Y n , Z ) → H ( X n , Z ) forgets that a class is relative (has compact support). Thus the abovediscussion shows that H ( X n , Y n , Z ) ∼ = H ( X n , Z ), equipped with the intersection form,is the root lattice of A n − , while H ( X n , Z ) is the dual weight lattice.Notice that in the simple case with n = 2, where X = X ∼ = T ∗ S , the generatorof H ( X , Y , Z ) ∼ = Z may be taken to be a compactly supported four-form that hasintegral one over the fibre (the Thom class of the bundle T ∗ S ).We may now compute the cohomology of Y n = ∂ X n using the long exact sequence forthe pair ( X n , Y n ). Since the cohomology groups of both X n and ( X n , Y n ) vanish in all50egrees other than the top, middle and bottom, it follows that most of the cohomologyof Y n is also trivial. In fact the only non-trivial cohomology group is H ( Y n , Z ), whicharises from the sequence · · · −→ H ( X n , Y n , Z ) f −→ H ( X n , Z ) −→ H ( Y n , Z ) −→ H ( X n , Y n , Z ) ∼ = 0 . (A.106)This implies that H ( Y n , Z ) ∼ = H ( X n , Z ) /f ( H ( X n , Y n , Z )) ∼ = Z n , where the last iso-morphism follows from the above description of the cohomology groups in terms ofthe root and weight lattices of A n − . Of course, by Poincar´e duality we also have H ( Y n , Z ) ∼ = Z n .In the special case that n = 2, of main interest in the text, the long exact homologysequence implies that we may take the boundary S of any fibre S = ∂ R of T ∗ S asthe generator of H ( Y , Z ) ∼ = Z . Equivalently, viewing Y as an S bundle over S , acopy of the fibre at any point on the base generates this third homology group.Next we introduce the free circle action on Y n by U (1) b ∼ = SO (2) diag ⊂ SO (4),where SO (4) acts on the coordinates z i , i = 1 , . . . , , in the vector representation. Thequotient M n = Y n /U (1) b is then a smooth compact six-manifold. The cohomology ofthis space may be deduced from the Gysin sequence for the circle fibration of Y n over M n : · · · −→ H i − ( M n , Z ) ∪ c −→ H i ( M n , Z ) −→ H i ( Y n , Z ) −→ H i − ( M n , Z ) −→ · · · . (A.107)It is straightforward to derive this sequence from the long exact sequence for the totalspace L of the complex line bundle over M n associated to the U (1) b circle bundle:note that L has boundary Y n , and base M n . One needs to combine this sequencewith the Thom isomorphism – this is precisely where the cup product with c = c ( L )comes from above, since for a complex line bundle c is equal to the Euler class of theunderlying rank 2 real vector bundle. The last map in the Gysin sequence (A.107) isjust pull-back from M n to Y n .Using the sequence (A.107), together with the known cohomology of Y n computedabove, we may compute the cohomology (and properties of the cohomology ring) of M n . Since H ( Y n , Z ) ∼ = H ( Y n , Z ) ∼ = 0, it follows immediately from i = 2 in (A.107)that c ≡ Ω is the generator of H ( M n , Z ) ∼ = Z . Here the notation Ω was introducedin the main text just before equation (3.33). Similarly, H ( Y n , Z ) ∼ = 0 implies that51 ( M n , Z ) ∼ = 0. Then i = 4 above implies Z n ∼ = H ( Y n , Z ) ∼ = H ( M n , Z ) / [ H ( M n , Z ) ∪ c ]. Now, H ( M n , Z ) ∼ = H ( M n , Z ), so the free part of H ( M n , Z ) is Z ∼ = H ( M n , Z ) bythe Universal Coefficient Theorem. Moreover, the torsion in H ( M n , Z ) is the torsionin H ( M n , Z ), but this is Poincar´e dual to H ( M n , Z ) ∼ = 0. Thus H ( M n , Z ) ∼ = Z , andthe Gysin sequence thus tells us that the square of the generator of H ( M n , Z ) is n times the generator of H ( M n , Z ). We may equivalently state this as Z Σ Ω ∪ Ω = n , (A.108)where Σ denotes the generator of H ( M n , Z ), again as in the main text. The result(A.108) follows from Poincar´e duality, and the last map in the Gysin sequence that sayscupping H ( M n , Z ) with c = Ω (which is Poincar´e dual to Σ ) maps the generatorof H ( M n , Z ) to the generator of H ( M n , Z ) ∼ = Z . Notice that M n then has the samecohomology groups as CP (where M ∼ = CP ), but that the cohomology ring dependson n via the above calculation.We may now compute the cohomology of the quotient Y n / Z k . This is also a smoothseven-manifold, where we take Z k ⊂ U (1) b . This immediately gives π ( Y n / Z k ) ∼ = H ( Y n / Z k , Z ) ∼ = Z k . The Gysin sequence (A.107), with Y n / Z k in place of Y n , nowhas c = k Ω . Precisely as we argued above, this implies the important result that H ( Y n / Z k , Z ) ∼ = H ( M n , Z ) / [ H ( M n , Z ) ∪ k Ω ] ∼ = Z nk . Of course, by Poincar´e dualityalso H ( Y n / Z k , Z ) ∼ = Z nk . Indeed, the Poincar´e dual sequence implies that the genera-tor Σ of H ( M n , Z ) ∼ = Z lifts to the generator Σ of H ( Y n / Z k , Z ) ∼ = Z nk , where Σ isthe total space of the circle bundle over a representative of Σ . This was used at theend of section 3.5.Finally, recall that in the special case of n = 2 the generator of H ( Y , Z ) ∼ = Z canbe taken to be a copy of the fibre S in the fibration S ֒ → Y → S . The fibres overthe poles p N , p S of the S are mapped into themselves under Z k , with the Hopf actionof Z k on S giving the quotient S / Z k . It then follows from the last paragraph thatthis Lens space S / Z k ∼ = Σ generates H ( Y / Z k , Z ) ∼ = Z k .52 The Stenzel metric
In this appendix we review the construction of the Stenzel metric on
X ∼ = T ∗ S . Thedeformed quadric X is defined as X i =0 z i = γ , (B.109)and the Stenzel metric on this may be written by introducing left-invariant one-forms L AB on SO (5), A, B = 1 , . . . ,
5, satisfying d L AB = L AC ∧ L CB . We split A = (1 , , i ),with i = 1 , ,
3, where the L ij are left-invariant one-forms for SO (3), and define σ i = L i , ˜ σ i = L i , ν = L . (B.110)These are one-forms on the coset space V , = SO (5) /SO (3). The metric on (B.109)is then [15] d s = c d r + c ν + a σ i + b ˜ σ i . (B.111)It is useful to introduce the orthonormal frame e = c d r , e ˜0 = cν , e i = aσ i , e ˜ i = b ˜ σ i . (B.112)A holomorphic frame is provided by ǫ = − e + i e ˜0 , ǫ i = e i + i e ˜ i . (B.113)In this frame, we take the K¨ahler form J and holomorphic (4 , J = i2 ǫ α ∧ ¯ ǫ ¯ α , Ω = ǫ ∧ ǫ ∧ ǫ ∧ ǫ . (B.114)Thus these automatically satisfy the SU (4)-structure algebraic relations J ∧ Ω = 0, J = Ω ∧ ¯Ω = − e . A Ricci-flat K¨ahler metric requires d J = 0 = dΩ. It isstraightforward to check that d J = 0 is equivalent to the ordinary differential equation(ODE) ( ab ) ′ = c , (B.115)53here a prime denotes differentiation with respect to r , while imposing dΩ = 0 isequivalent to the four ODEs 3 a ′ a + c ′ c − ba = 0 , b ′ b + c ′ c − ab = 0 , a ′ a + b ′ b + c ′ c − ba − ab = 0 , b ′ b + a ′ a + c ′ c − ab − ba = 0 . (B.116)Although this naively looks overdetermined, it is simple to check by taking linearcombinations that these five ODEs are equivalent to the three ODEs a ′ a = b + c − a ab ,b ′ b = a + c − b ab ,c ′ c = 3( a + b − c )2 ab . (B.117)This is the same system of equations that were presented in [15], although in thelatter reference they were derived by first finding the second order Einstein equations,and then constructing a superpotential. Here we have derived them directly from theRicci-flat K¨ahler conditions. A solution to these equations, which is a smooth completemetric on X = T ∗ S , was found by Stenzel [54]. This is the solution written in (5.61). C A different reduction to Type IIA
In sections 3.2 and 3.5 we considered reducing M-theory on R , × X n / Z k with N spacefilling M2-branes, or its near-horizon limit AdS × Y n / Z k , along U (1) b to TypeIIA string theory. Recall here that X n admits a Ricci-flat K¨ahler cone metric only for n = 1 and n = 2. In the case n = 2, one problem with this Type IIA reduction is thatas soon as one deforms the AdS × Y / Z k solution to the R , × X / Z k solution, thereduction along U (1) b is no longer well-behaved. Specifically, the U (1) b action fixesthe north and south poles of the S zero-section of X ≡ X ∼ = T ∗ S ; since these arecodimension eight, there is no simple interpretation of the resulting singularity in thedilaton in Type IIA string theory. Thus the Type IIA supergravity solution cannotbe trusted in the IR region near to the S at r = 0. However, there is a different is well-behaved. We briefly describe this here, leaving amore thorough investigation for future work.Recall that in section 4.2 we introduced a different U (1) ≡ U (1) action on X n .If we regard X n as being defined by the hypersurface equation (2.9), the coordinates( A , A , B , B , z = [ s ( n + 1)] /n Φ ) have charges (1 , , − , ,
0) under U (1) . In fact,we may deform X n to X n given by (A.105), so that U (1) also acts on the smoothmanifold X n . Of course, to obtain a solution to eleven-dimensional supergravity, weshould equip X n with a Calabi-Yau metric. For n = 1, n = 2, we may use completeasymptotically conical Calabi-Yau metrics (the flat metric on X ∼ = C ; the Stenzelmetric on X ∼ = T ∗ S ). These are the metrics relevant for application to the AdS/CFTcorrespondence. Such metrics do not exist for n >
2, in which case the reader canimagine that (A.105) is a local model in a compact
Calabi-Yau manifold. Yau’s theoremwill then give a Ricci-flat K¨ahler metric on this space which is incomplete at theboundary. In any case, the precise details of the metric will not be important in whatfollows.Consider reduction of M-theory on R , × X n , with N spacefilling M2-branes, along U (1) . The fixed point set is codimension four, namely { A = B = 0 } , which cuts outthe locus n Y γ =1 ( z − a γ ) + A B = 0 . (C.118)This is the deformation of the A n − singularity: it has n − S γ , definedsimilarly to the four-spheres S γ in appendix A, that intersect according to the rootlattice of A n − = SU ( n ). This becomes a D6-brane locus when we reduce to Type IIA.Indeed, the Type IIA spacetime is flat, since X n /U (1) ∼ = R . To see this, note that X n / C ∗ is described by z + n Y γ =1 ( z − a γ ) + A B = 0 . (C.119)where z = A B . This is simply C . The quotient space is thus diffeomorphic to R ∼ = R × C , where R is spanned by | A | − | B | , which one can think of as themoment map for U (1) , and C is spanned by ( A , B , z ). The fixed point locus isthus at the origin of R , and cuts out the hypersurface (C.118) in the C part.The reduction of R , ×X n along U (1) is thus the flat spacetime R , = R , × R × C ,with N spacefilling D2-branes and a single spacefilling D6-brane sitting at the origin55f R and wrapping the divisor (C.118) in C . Notice that this description gives thecorrect amount of supersymmetry, since a D-brane wrapped on a divisor in a three-foldpreserves four supercharges, or N = 2 supersymmetry in d = 3.There are n − X n , and the quantized G -flux through the generators S γ defined in appendix A gives 1(2 πl p ) Z S γ G = M γ ∈ Z . (C.120)In the Type IIA reduction considered in this section, this is dual to adding M γ unitsof worldvolume gauge field flux on the D6-brane through the two-sphere S γ in thedeformed A n − singularity (C.118). A general discussion of this may be found in [55].Thus 12 πl s g s Z S γ F = M γ , (C.121)where F is the U (1) gauge field on the D6-brane.In the limit where a γ →
0, which is the hypersurface singularity X n , the D6-brane iswrapped on R , ×A n − (we emphasize that the spacetime is flat Minkowski spacetime).In particular, for n = 2 we have an A singularity, although for n > A quiver in section 2 is not related to this A singularity in the TypeIIA reduction on U (1) . Indeed, since we are reducing on a different circle, one expectsthe effective gauge theory derived from the brane configuration described above to be mirror to the gauge theory in section 2, which we derived from the Type IIA reductionon U (1) b in section 3.5.We may also consider taking the Z k quotient along U (1) b . The charges of the coor-dinates ( A , A , B , B , z ) under U (1) b are (1 , , − , − , R × C , spanned by the moment map | A | − | B | and ( A , B , z ), re-spectively, U (1) b acts with charges (1 , − ,
0) on C . Thus the Z k quotient along U (1) b leads to a Z k singularity in spacetime , or more precisely an A k − singularity. Thiswould usually lead to an SU ( k ) gauge symmetry in the transverse six-dimensionalspace. Contrast this with the A n − singularity on which the D6-brane is wrapped.Finally, notice that we may perform a T-duality along the U (1) which acts withcharges (1 , −
1) on the coordinates ( A , B ). This gives a Type IIB brane set-up wherethe spacetime is R , × R × S × R , with N spacefilling D3-branes wrapping the S circle (that arises from the T-duality). Here R arises as R = R × C , where R isspanned by the moment map | A | − | B | , and C is spanned by ( z , A B ). Since56he fixed point locus is { A = B = 0 } , which is a copy of R , × R × C in theIIA spacetime (with C spanned by the coordinate z ), on T-dualizing this becomes alinearly embedded spacefilling NS5-brane. More precisely, the NS5-brane wraps the R direction, sits at a point in S , and wraps the copy of C ⊂ R spanned by the coordinate z . When we divide by Z k ⊂ U (1) b , the fixed locus is precisely the A k − singularity,and we thus obtain k linearly embedded spacefilling NS5-branes in the Type IIB dual.The spacefilling D6-brane wrapped on the deformation of the A n − singularity becomesa spacefilling D5-brane wrapped on a non-linearly embedded copy of R in R . 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