Contents

July 12, 2019

Baryonic symmetries and M5 branes in

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Nessi Benishti,    Diego Rodríguez-Gómez,   and   James Sparks

1: Rudolf Peierls Centre for Theoretical Physics,

University of Oxford,

1 Keble Road, Oxford OX1 3NP, U.K.

2: Queen Mary, University of London,

Mile End Road, London E1 4NS, U.K.

3: Mathematical Institute, University of Oxford,

24-29 St Giles’, Oxford OX1 3LB, U.K.

We study symmetries dual to Betti multiplets in the correspondence for M2 branes at Calabi-Yau four-fold singularities. Analysis of the boundary conditions for vector fields in allows for a choice where wrapped M5 brane states carrying non-zero charge under such symmetries can be considered. We begin by focusing on isolated toric singularities without vanishing six-cycles, and study in detail the cone over . The boundary conditions considered are dual to a CFT where the gauge group is . We find agreement between the spectrum of gauge-invariant baryonic-type operators in this theory and wrapped M5 brane states. Moreover, the physics of vacua in which these symmetries are spontaneously broken precisely matches a dual gravity analysis involving resolutions of the singularity, where we are able to match condensates of the baryonic operators, Goldstone bosons and global strings. We also argue more generally that theories where the resolutions have six-cycles are expected to receive non-perturbative corrections from M5 brane instantons. We give a general formula relating the instanton action to normalizable harmonic two-forms, and compute it explicitly for the example. The holographic interpretation of such instantons is currently unclear.

## 1 Introduction

Over the last two years there have been major advances towards understanding the duality. Elaborating on [1, 2], Aharony, Bergman, Jafferis and Maldacena [3] proposed a theory conjectured to be dual to M2 branes probing a singularity, where acts with weights on the coordinates of . This low energy theory on the worldvolume of coincident M2 branes is a quiver Chern-Simons (CS) theory, with a marginal quartic superpotential whose coefficient is related by the high degree of SUSY to the CS coupling. Indeed, for generic CS coupling the theory enjoys SUSY, and as such possesses an symmetry which is manifest in the potential [4]. The theory is then automatically conformal at the quantum level. For the SUSY is enhanced to . In field theory it has been argued [3] that this enhancement is due to quantum effects where ’t Hooft monopole operators play a key rôle. Indeed, the ABJM theory has just the right structure [5] for these monopole operators to have appropriate quantum numbers that then allow for such a symmetry enhancement.

Motivated by this progress in understanding the maximally SUSY case, it is natural to consider M2 branes moving in less symmetric spaces, leading to versions of the duality with reduced SUSY. Inspired by ABJM [3], the theories considered are quiver CS (QCS) theories with bifundamental matter. The rôle of the CS levels is far from trivial, and it has been argued in [6, 7] that the sum corresponds to the Type IIA SUGRA Romans mass parameter. In this paper we will focus entirely on the case in which the CS levels sum to zero; the Romans mass then vanishes and the system admits an M-theory lift.

Since the kinetic terms for the gauge fields are given by the CS action, the only classically dimensionful parameters are the superpotential couplings. Strong gauge dynamics is then conjectured to drive the theory to a superconformal IR fixed point. In [8, 9, 10], a general analysis of the moduli spaces of such superconformal gauge theories was presented. In particular, it is crucial that the CS levels sum to zero if there is to be a so-called geometric branch of the moduli space which is a Calabi-Yau four-fold cone, where the branes are interpreted as moving. In parallel to the ABJM case, multiplying the vector of CS levels by an integer orbifolds this moduli space in a certain way. More precisely, the theory with has a (abelian) moduli space which is a quotient of the moduli space of the theory with CS levels . Generically this group will not act freely away from the tip of the cone. In this case one might expect additional gauge symmetries at such fixed points. This has recently motivated [11, 12] the consideration of dual field theories which involve fundamental, as well as bifundamental, matter. It is however fair to say that, at present, there is no comprehensive understanding of these constructions.

On general grounds, the presence of global symmetries is of great help in classifying the spectrum of a gauge theory. One particularly important such global symmetry is the R-symmetry. In three dimensions a theory preserving supersymmetries admits the action of an R-symmetry. Thus the existence of a non-trivial R-symmetry, which can then provide important constraints on the dynamics, requires that we focus on , implying there is at least a . In particular it then follows that, assuming the theory flows to an IR superconformal fixed point, the scaling dimensions of chiral primary operators coincide with their R-charges. We note that, generically, the theories considered have classically irrelevant superpotentials. Strong gauge dynamics is required to give large anomalous dimensions, thus making it possible to reach a non-trivial IR fixed point. However, in three dimensions there are few independent field theory checks on the existence of such a fixed point. For example, there is no useful analogous version of -maximization [13], which for four-dimensional theories allows one to determine the R-charge in the superconformal algebra at the IR fixed point. This places the conjectured dualities on a much weaker footing than their four-dimensional cousins in Type IIB string theory.

The QCS theories that we consider are expected to be dual to M2 branes moving in a Calabi-Yau four-fold cone over a seven-dimensional Sasaki-Einstein base , thus giving rise to an near horizon dual geometry. Such Sasaki-Einstein manifolds will typically have non-trivial topology, implying the existence of Kaluza-Klein (KK) modes obtained by reduction of SUGRA fields along the corresponding homology cycles. Of particular interest are five-cycles, on which one can reduce the M-theory six-form potential to obtain vector fields in . These vector fields are part of short multiplets of the KK reduction on , known as Betti multiplets [14, 15] (for a discussion relevant to the cases we will consider, see also [16, 17]). In analogy with the Type IIB case, where these symmetries are well-known to correspond to global baryonic symmetries [18], we will sometimes employ the same terminology here and refer to these as baryonic s.

In this paper we set out to study the above symmetries in the correspondence. In the rather better-understood correspondence in Type IIB string theory, from the field theory point of view these baryonic symmetries appear as non-anomalous combinations of the diagonal factors inside the gauge groups.111For a complete discussion of the Type IIB case we refer to e.g. [19, 20, 21]. The key point is that, in four dimensions, abelian gauge fields are IR free and thus become global symmetries in the IR. However, this is no longer true in three dimensions, thus raising the question of the fate of these abelian symmetries. From the gravity perspective, in the dual the vector fields admit two admissible fall-offs at the boundary of [22, 23]. This is in contrast to the case where only one of them, that which leads to the interpretation as dual to a global current, is allowed. That the two behaviours are permitted implies that the corresponding boundary symmetries remain either gauged or ungauged, respectively, defining in each case a different boundary CFT. This issue is closely related to the gauge groups being either or in the case at hand. From the point of view of the QCS theory with gauge groups, at lowest CS level there is no real distinction between and gauge groups [3, 24, 25] . Therefore the discussion in [22] can be applied to the abelian part of the symmetry. In this way it is possible to connect the and the theories in a rather precise manner, while keeping track of the corresponding action on the gravity side, which amounts to selecting one particular fall-off for the vector fields in . This provides motivation to look at the version of the theory as dual to a particular choice of boundary conditions in the dual gravity picture.

In the first part of this paper we focus on the simplest class of examples, namely isolated toric Calabi-Yau four-fold singularities with no vanishing six-cycles (no exceptional divisors in a crepant resolution). These are discussed in more detail in §A. In particular we study in detail the example of . A dual QCS field theory was proposed for this singularity in [26], and further studied in [27] where the non-abelian chiral ring of the theory (at large ) was shown to precisely match the coordinate ring of the variety. Motivated by the analysis of the behaviour of gauge fields in , we will choose boundary conditions where the Betti multiplets are dual to global symmetries. This amounts to focusing on a certain version of the theory with gauge group . On the other hand, gauge fields in can have a priori both electric sources, corresponding to wrapped M5 branes, and magnetic sources, corresponding to wrapped M2 branes. It turns out that the boundary conditions necessary to define the correspondence allow for just one of the two types at a time [22]. In particular, the chosen quantization allows only for electric sources; that is, wrapped supersymmetric M5 branes. In turn, these correspond to baryonic operators [28] in the field theory that are charged under the global symmetries. We will analyse this correspondence in detail, finding the expected agreement.

On the other hand, magnetic sources correspond to M2 branes [28]. While in the geometry these wrap non-supersymmetric cycles, we can also consider resolutions of the corresponding cone where there are supersymmetric wrapped M2 branes. Along the lines of [29, 30], we will identify the relevant operator, responsible for the resolution, which is acquiring a VEV. Very much as in reference [30], it is possible to find an interpretation of these solutions as spontaneous symmetry breaking (SSB) through the explicit appearance of a Goldstone boson in the SUGRA dual.

A natural next step is to enlarge the class of singularities under consideration by allowing dual geometries with exceptional six-cycles. One such example is a orbifold of known as . A dual field theory candidate has been proposed in [27, 31, 32]. Further tests of this theory were performed in [27], where it was shown that its chiral ring matches the gravity computation at large . The interpretation of such six-cycles is somewhat obscure holographically. Indeed, such six-cycles, when resolved, can support M5 brane instantons leading to non-perturbative corrections [33]. In the second part of this paper we set up the study of such corrections by finding a general expression for the Euclidean action of such branes in terms of normalizable harmonic two-forms, and compute this explicitly for . We leave a full understanding of such non-perturbative effects from the gauge theory point of view for future work.

The organization of this paper is as follows. In §2 we review the Freund-Rubin-type solutions which are eleven-dimensional backgrounds. We then turn to KK reduction of the SUGRA six-form potential on five-cycles in , leading to the Betti multiplets of interest. General analysis of gauge fields in shows that two possible fall-offs are admissible. We then review the construction in [22] relating these different boundary conditions for a single abelian gauge field in to the action of . In §3 we turn in more detail to the field theory description. We start by reviewing general aspects of QCS theories that have appeared in the literature, before turning in §3.2 to the example of interest. We then propose a set of boundary conditions dual to the theory. We identify the ungauged s via the electric M5 branes wrapping holomorphic divisors in the geometry. In §4 we turn to the spontaneous breaking of these baryonic symmetries. We compute on the gravity side the baryonic condensate and identify the Goldstone boson of the SSB. In §5 we initiate the study of exceptional six-cycles. We compute the warped volume of a Euclidean brane in the resolved geometry. By extending our results on warped volumes to arbitrary geometries, both for the baryonic condensate and the Euclidean brane, we find general formulae for such warped volumes. We end with some concluding comments in §6. Finally, a number of relevant calculations and formulae are collected in the appendices.

Note added: as this paper was being finalized the preprint [70] appeared, which has partial overlap with our results.

## 2 AdS4 backgrounds and abelian symmetries

We begin by reviewing general properties of Freund-Rubin backgrounds, and also introduce the and examples of main interest. KK reduction of the M-theory potentials on topologically non-trivial cycles leads to gauge symmetries in . We review their dynamics in the context and the sources allowed, depending on the chosen quantization. Of central relevance for our purposes will be wrapped supersymmetric M5 branes.

### 2.1 Freund-Rubin solutions

The backgrounds of interest are of Freund-Rubin type, with eleven-dimensional metric and four-form given by

 ds211 = R2(14ds2(AdS4)+ds2(Y)) , (2.1) G = 38R3Vol(AdS4) .

Here the metric is normalized so that . The Einstein equations imply that is an Einstein manifold of positive Ricci curvature, with metric normalized so that . The flux quantization condition

 1(2πℓp)6∫Y⋆11G=N∈Z , (2.2)

then leads to the relation

 R=2πℓp(N6vol(Y))1/6 , (2.3)

where denotes the eleven-dimensional Planck length.

As is well-known, such solutions arise as the near-horizon limit of M2 branes placed at the tip of the Ricci-flat cone

 ds2(C(Y))=dr2+r2ds2(Y) . (2.4)

More precisely, the eleven-dimensional solution is

 ds211 = h−2/3ds2(R1,2)+h1/3ds2(X) , (2.5) G = d3x∧dh−1 ,

where in the case at hand we take the eight-manifold with conical metric (2.4). Placing Minkowski space-filling M2 branes at leads, after including their gravitational back-reaction, to the warp factor

 h=1+R6r6 . (2.6)

In the near-horizon limit, near to , the background (2.5) approaches the background (2.1). In fact the warp factor is precisely the background in a Poincaré slicing. More precisely, writing

 z=R2r2 ,ds2(AdS4)=z−2(dz2+ds2(R1,2)) , (2.7)

leads to the metric (2.1).

We shall be interested in solutions of this form preserving supersymmetry in . The well-known result [34] is summarized in Table 1. As mentioned in the introduction, in general supersymmetries leads to the R-symmetry group , and thus supersymmetry provides a strong constraint on the spectrum only for . We hence restrict attention to the Sasaki-Einstein case, which includes the geometry as a special case. It is then equivalent to say that the cone metric on is Kähler as well as as Ricci-flat, i.e. Calabi-Yau. Geometries with supersymmetries are necessarily quotients of .

Only a decade ago the only known examples of such Sasaki-Einstein seven-manifolds were homogeneous spaces. Since then there has been dramatic progress. 3-Sasakian manifolds, with , may be constructed via an analogue of the hyperKähler quotient, leading to rich infinite classes of examples [35]. For supersymmetry one could take to be one of the explicit manifolds constructed in [36], and further studied in [37, 38], or any of their subsequent generalizations. These examples are all toric, meaning that the isometry group contains as a subgroup. In fact, toric Sasaki-Einstein manifolds are now completely classified thanks to the general existence and uniqueness result in [39]. At the other extreme, there are also non-explicit metrics in which is the only isometry [35].

However, for our purposes it will be sufficient to focus on two specific homogeneous examples, namely and , with being along the R-symmetry of . These will turn out to be simple enough so that everything can be computed explicitly, and yet at the same time we shall argue that many of the features seen in these cases hold also for the more general geometries mentioned above. In both cases the isometry group is , and in local coordinates the explicit metrics are

 ds2=116(dψ+3∑i=1cosθidϕi)2+183∑i=1(dθ2i+sin2θidϕ2i) . (2.8)

Here are standard coordinates on three copies of , , and has period for and period for . The two Killing spinors are charged under , which is dual to the symmetry. The metric (2.8) shows very explicitly the regular structure of a bundle over the standard Kähler-Einstein metric on , where is the fibre coordinate and the Chern numbers are and respectively. These are hence natural generalizations222The other natural such generalization is the homogeneous space , which has been studied in detail in [40]. to seven dimensions of the and manifolds.

### 2.2 C-field modes

One might wonder whether it is possible to turn on an internal -flux on , in addition to the -field in (2.1), and still preserve supersymmetry, i.e.

In fact necessarily . This follows from the results of [41]: for any warped Calabi-Yau four-fold background with metric of the form (2.5), one can turn on a -field on without changing the Calabi-Yau metric on only if is self-dual. But for a cone, with a pull-back from the base , this obviously implies that .

However, more precisely the -field in M-theory determines a class333This is true since the membrane global anomaly described in [42] is always zero on a seven-manifold that is spin. in . The differential form part of captures only the image of this in , and so still allows for a topologically non-trivial -field classified by the torsion part . This is also captured, up to gauge equivalence, by the holonomy of the corresponding flat -field through dual torsion three-cycles in . There are hence physically distinct Freund-Rubin backgrounds associated to the same geometry, which should thus correspond to physically inequivalent dual SCFTs. In a small number of examples with proposed Chern-Simons quiver duals, including the original ABJ(M) theory, different choices of this torsion -flux have been argued to be dual to changing the ranks in the quiver [40, 43]. However, the related Seiberg-like dualities are currently very poorly understood in examples without Hanany-Witten-type brane duals. In particular, for example, one can compute , implying there are two distinct M-theory backgrounds with the same geometry but different -fields. This is an important aspect of the duality that we shall not discuss any further in this paper.

More straightforwardly, if one has three-cycles in then one can also turn on a closed three-form with non-zero periods through these cycles. Including large gauge transformations, this gives a space of such flat -fields. Since these are continuously connected to each other they would be dual to marginal deformations in the dual field theory. Indeed, the harmonic three-forms on a Sasaki-Einstein seven-manifold are in fact paired by an almost complex structure [44] and thus is always even, allowing these to pair naturally into complex parameters as required by supersymmetry. However, for the class of toric singularities studied in this paper, including and , it is straightforward444There are, however, examples: the Calabi-Yau four-fold hypersurfaces , where , are known to have Calabi-Yau cone metrics, and these have , , respectively [44]. to show that and there are hence no such marginal deformations associated to the -field.

Finally, since for any positively curved Einstein seven-manifold, there are never periods of the dual potential through six-cycles in .

### 2.3 Baryonic symmetries and wrapped branes

Of central interest in this paper will be symmetries associated to the topology of , and the corresponding charged BPS states associated to wrapped M branes. By analogy with the corresponding situation in in Type IIB string theory, we shall refer to these symmetries as baryonic symmetries; the name will turn out to be justified.

Denote by the second Betti number of . By Poincare duality we have . Let be a set of dual harmonic five-forms with integer periods. Then for the Freund-Rubin background we may write the KK ansatz

 δC6=2πT5b2(Y)∑I=1AI∧αI , (2.10)

where is the M5 brane tension. This gives rise to massless gauge fields in . For a supersymmetric theory these gauge fields of course sit in certain multiplets, known as Betti multiplets. See, for example, [14, 15, 16, 17].

#### 2.3.1 Vector fields in AdS4, boundary conditions and dual CFTs

The duality requires specifying the boundary conditions for the fluctuating fields in . In particular, vector fields in admit different sets of boundary conditions [22, 23] leading to different boundary CFT´s. In order to see this, let us consider a vector field in . Using the straightforward generalization to of the coordinates in (2.7), in the gauge the bulk equations of motion set

 Aμ=aμ+jμzd−2 , (2.11)

where satisfy the free Maxwell equation in Lorentz gauge in the Minkowski space. It is not hard to see that in both behaviours have finite action, and thus can be used to define a consistent duality.

Let us now concentrate on the case of interest , where both quantizations are allowed. In order to have a well-defined variational problem for the gauge field in we should be careful with the boundary terms when varying the action. In general, we have

 δS=∫{∂√detgL∂AM−∂N∂√detgL∂∂NAM}δAM+∂N{∂√detgL∂∂NAMδAM} . (2.12)

The bulk term gives the equations of motion whose solution behaves as (2.11). In turn, the boundary term can be seen to reduce to

 δSB=−12∫Boundaryjμδaμd3x . (2.13)

Therefore, in order to have a well-posed variational problem, we need to demand ; that is, we need to impose boundary conditions where is fixed in the boundary.

On the other hand, since in both behaviours for the gauge field have finite action, we can consider adding suitable boundary terms such that the action becomes [23]

 S=14∫√detgFABFAB+12∫Boundary√detgAμFrμ|Boundaryd3x. (2.14)

The boundary term is now

 δSB=12∫Boundaryaμδjμd3x , (2.15)

so that we need to impose the boundary condition ; that is, fix the boundary value of .

Defining and , we have

 Bμ=ϵμνρ∂νaρ+ϵμνρ∂νjρz,Eμ=jμz2 . (2.16)

The two sets of boundary conditions then correspond to either setting while leaving unrestricted, or setting while leaving unrestricted.

At this point we note that are naturally identified, respectively, with a dynamical gauge field and a global current in the boundary. In accordance with this identification, eq. (2.11) and the usual prescription shows each field to have the correct scaling dimension for this interpretation: for a gauge field , while for a global current . Therefore, the quantization is dual to a boundary CFT where the gauge field is dynamical; while the quantization is dual to a boundary CFT where the is ungauged and is instead a global symmetry. Furthermore, as discussed in [18] for the scalar counterpart, once the improved action is taken into account the two quantizations are Legendre transformations of one another [70], as can be seen by e.g. computing the free energy in each case.

One can consider electric-magnetic duality in the bulk theory, which exchanges thus exchanging the two boundary conditions for the gauge field quantization. This action translates in the boundary theory into the so-called operation [22]. This is an operation on three-dimensional CFTs with a global symmetry, taking one such CFT to another. In addition, it is possible to construct a operation, which amounts, from the bulk perspective, to a shift of the bulk -angle by . Following [22], we can be more precise in defining these actions in the boundary CFT. Starting with a three-dimensional CFT with a global current , one can couple this global current to a background gauge field resulting in the action . The operation then promotes to a dynamical gauge field and adds a BF coupling of to a new background field , while the operation instead adds a CS term for the background gauge field :

 (2.17)

As shown in [22], these two operations generate the group .555Even though we are explicitly discussing the effect of on the vector fields, since these are part of a whole Betti multiplet we expect a similar action on the other fields of the multiplet. We leave this investigation for future work. In turn, as discussed above, the and operations have the bulk interpretation of exchanging and shifting the bulk -angle by , respectively. It is important to stress that these actions on the bulk theory change the boundary conditions. Because of this, the dual CFTs living on the boundary are different.

#### 2.3.2 Boundary conditions and sources for gauge fields: M5 branes in toric manifolds

We are interested in gauge symmetries in associated to the topology of ; that is, arising from KK reductions as in (2.10). All Kaluza-Klein modes, and hence their dual operators, carry zero charge under these symmetries. However, there are operators associated to wrapped M branes that do carry charge under this group. In particular, an M5 brane wrapped on a five-manifold , such that the cone is a complex divisor in the Kähler cone , is supersymmetric and leads to a BPS particle propagating in . Since the M5 brane is a source for , this particle is electrically charged under the massless gauge fields . One might also consider M2 branes wrapped on two-cycles in . However, such wrapped M2 branes are supersymmetric only if the cone over the two-submanifold is calibrated in the Calabi-Yau cone, and there are no such calibrating three-forms. Thus these particles, although topologically stable, are not BPS. They are magnetically charged under the gauge fields in [28].

As discussed above, the duality instructs us to choose, for each gauge field, a set of boundary conditions where either or vanishes. Clearly, only the latter possibility allows for the existence of the SUSY electric M5 branes, otherwise forbidden by the boundary conditions. In turn, this quantization leaves, in the boundary theory, the symmetry as a global symmetry. Therefore, in this case we should expect to find operators in the field theory that are charged under the global baryonic symmetries and dual to the M5 brane states. We turn to this point in the next section. We note that, with this choice of boundary condition, the rôle of the Betti multiplets is very similar to their counterparts, giving rise to global baryonic symmetries in the boundary theory, and hence motivating the use of the same name in the case at hand.

For toric manifolds there is a canonical set of such wrapped M5 brane states, where are taken to be the toric divisors. Each such state leads to a corresponding dual chiral primary operator that is charged under the global symmetries and will also have definite charge under the flavour group dual to the isometries of . We refer the reader to the standard literature for a thorough introduction to toric geometry. However, the basic idea is simple to state. The cone fibres over a polyhedral cone in with generic fibre . This polyhedral cone is by definition a convex set of the form , where are integer vectors. This set of vectors is precisely the set of charge vectors specifying the subgroups of that have complex codimension one fixed point sets. These fixed point sets are, by definition, the toric divisors referred to above. The Calabi-Yau condition implies that, with a suitable choice of basis, we can write , with . If we plot these latter points in and take their convex hull, we obtain the toric diagram.

For the example the toric divisors are given by taking or , for any , which are 6 five-manifolds in . The toric diagram for is shown in Figure 1, where one sees clearly these 6 toric divisors as the 6 external vertices. Notice that for the full isometry group may be used to rotate into , specifically using the th copy of in the isometry group. In fact these two five-manifolds are two points in an family of such five-manifolds related via the isometry group. Similar comments apply also to .

## 3 Baryonic symmetries in QCS theories

In the previous section we discussed the rôle of vector fields in . In particular, we have shown that there is a choice of boundary conditions where the Betti multiplets corresponding to (2.10) are dual to global currents in the boundary theory. From the bulk perspective, this translates into the possibility of having electric M5 brane states in the theory, in a consistent manner. On general grounds, we expect these states to be dual to certain operators in the boundary theory charged under the global . In this section we turn to a more precise field theoretic description of this. We begin with a brief review of the theories considered in the literature, before turning to our example and considering the rôle of the abelian symmetries in this case.

### 3.1 U(n) QCS theories

Let us start by considering the theories. The Lagrangian, in superspace notation, for a theory containing an arbitrary number of bifundamentals in the representation under the -th gauge groups and a choice of superpotential , reads

 L= ∫d4θTr⎡⎢⎣∑XabX†abe−VaXabeVb+G∑a=1ka2π1∫0dtVa¯Dα(etVaDαe−tVa)⎤⎥⎦ (3.1) +∫d2θW(Xab)+c.c. .

Here are the CS levels for the vector multiplet . For future convenience we define .

The classical vacuum moduli space (VMS) is determined in general by the following equations [9, 10]

 ∂XabW = 0 , μa:=−G∑b=1Xba†Xba+G∑c=1XacXac† = kaσa2π , σaXab−Xabσb = 0 , (3.2)

where is the scalar component of . Following [9], upon diagonalization of the fields using rotations, one can focus on the branch where , , so that the last equation is immediately satisfied.666We stress that there might be, and indeed even in the example there are, other branches of the moduli space where the condition for all is not met, and yet still the bosonic potential is minimized. Under the assumption that , the equations for the moment maps boil down to a system of independent equations for the bifundamental fields, analogous to D-term equations. Since for toric superpotentials the set of F-flat configurations, determining the so-called master space, is of dimension , upon imposing the D-terms and dividing by the associated gauge symmetries we have a moduli space where the M2 branes move.

However, due to the peculiarities of the CS kinetic terms, extra care has to be taken with the diagonal part of the gauge symmetry. At a generic point of the moduli space the gauge group is broken to copies of . The diagonal gauge field is completely decoupled from the matter fields, and only appears coupled to through

 S(BG)=k4πG∫(BG−1)μϵμνρ(GG)νρ . (3.3)

Since appears only through its field strength, it can be dualized into a scalar . Following the standard procedure, it is easy to see that integrating out sets

 BG−1=Gkdτ , (3.4)

such that the relevant part of the action becomes a total derivative

 (3.5)

Around a charge monopole in the diagonal gauge field we then have , so that must have period in order for the above phase to be unobservable [9]. Gauge transformations of then allow one to gauge-fix to a particular value via (3.4), but this still leaves a residual discrete set of gauge symmetries that leave this gauge choice invariant. The space of solutions to (3.2) is then quotiented by gauge transformations where the parameters satisfy , together with the residual discrete gauge transformations generated by for all . Altogether this leads to a quotient. We refer to [9] for further discussion, and to [27] for a discussion in the context of the theory in particular.

An alternative point of view has recently appeared in the literature [11, 12], in which the existence of two special monopole operators is noted. These monopole operators, which have charges respectively under each gauge group, are conjectured to satisfy a relation in the chiral ring of the form . In this approach the moduli space is defined as the chiral ring of the abelian theory enhanced by the operators , together with the constraint.

### 3.2 The C(Q111) theory

#### 3.2.1 The theory and its moduli space

A field theory candidate dual to M2 branes probing was proposed in [26] and further studied in [27]. The proposal in those references is a Chern-Simons gauge theory with CS levels , with matter content summarized by the quiver in Figure 2.

In addition, there is a superpotential given by

 W=Tr(C2B1A1B2C1A2−C2B1A2B2C1A1) . (3.6)

As expected for a field theory dual to point-like branes moving in , the moduli space contains a branch which is the symmetric product of copies of this conical singularity. To see this, let us begin with the abelian theory in which all the gauge groups are . As shown in [27], after integrating out the auxiliary scalar the geometric branch of the moduli space with is described by D-term equations. Recalling the special rôle played by , it is useful to introduce the following basis for the gauge fields:

 B3=A1+A2−A3−A4 ,B4=A1+A2+A3+A4 .

Then the two D-terms to impose are those for . In turn, the charge matrix is

 A1A2B1B2C1C2U(1)I1100−1−1U(1)II−1−11100U(1)B300−22−22U(1)B4000000 . (3.7)

Notice the appeareance of the global symmetry, under which the pairs , , transform as doublets under each of the respective factors.

Since for the abelian theory the superpotential is identically zero, one can determine the abelian moduli space by constructing the gauge-invariants with respect to the gauge transformations for . Borrowing the results from [27], for CS level these are

 w1=A1B2C1 ,w2=A2B1C2 ,w3=A1B1C2 ,w4=A2B2C1 ,w5=A1B1C1 ,w6=A2B1C1 ,w7=A1B2C2 ,w8=A2B2C2. (3.8)

One can then check explicitly that these satisfy the 9 relations defining as an affine variety:

 w1w2−w3w4=w1w2−w5w8=w1w2−w6w7=0 ,w1w3−w5w7=w1w6−w4w5=w1w8−w4w7=0 ,w2w4−w6w8=w2w5−w3w6=w2w7−w3w8=0 . (3.9)

This is an affine toric variety, with toric diagram given by Figure 1. Indeed, we also notice that for the abelian theory the description of the moduli space as a Kähler quotient of with coordinates is precisely the minimal gauged linear sigma model (GLSM) description. Thus the 6 toric divisors in Figure 1, discussed in §2.3, are defined by , , , .

For CS level one obtains an supersymmetric orbifold of . Notice that are invariant under this action, while and are rotated with equal and opposite phase. On the other hand, for the non-abelian theory with it was shown in [27] that for large , where the use of still poorly-understood monopole operators is evaded, upon using the F-terms of the full non-abelian superpotential (3.6) the chiral ring matches that expected for the corresponding orbifold. In this case the chiral primaries at the non-abelian level are just the usual gauge-invariants given by

 Tr(r∏a=1X±ia) ,whereX+i=AiC2B1 ,X−i=AiB2C1 . (3.10)

An important subtlety in this theory is that does not act freely on : it fixes two disjoint copies of inside , as explained in [11]. Indeed, using (3.7) one sees that the corresponding two cones are parametrized respectively by and , with in each case all other . Thus for the horizon has orbifold singularities in codimension four. This means that the SUGRA approximation cannot be trusted for . In fact these are singularities which can support “fractional” M2 branes wrapping the collapsed cycles, and one expects an gauge theory to be supported on these s. A different perspective can be obtained by interpreting as the M-theory circle and reducing to Type IIA. This results in D6 branes wrapping these two submanifolds. From now on we will therefore assume that .

#### 3.2.2 Gauged versus global abelian subgroups and SL(2,Z)

At the orbifold identification due to the CS terms is trivial. Indeed, in this case there is no real distinction between and gauge groups, as discussed in [3, 24, 25] for the ABJM theory and orbifolds of it. We shall argue that ungauging some of the s is dual to a particular choice of boundary conditions on the gravity side. That is, we apply the general discussion in §2.3 to the gauge fields, and argue that the associated symmetries are those in , for appropriate gauge group factors. This raises the important problem of how to identify the relevant two symmetries dual to the Betti multiplets in the QCS theory proposed above. The key is to recall that the boundary conditions which amount to ungauging these s in turn allow for the existence of supersymmetric M5 branes on the gravity side. As discussed in §2.3, from an algebro-geometric point of view the corresponding divisors are easy to identify. In turn we notice that, for the abelian theory, the fields are also the minimal GLSM coordinates. Setting each to zero therefore gives one of the 6 toric divisors that may be wrapped by an M5 brane. The charges of the resulting M5 brane states under are then the same as the charges of these fields under the we quotient by in forming the abelian moduli space – this was shown for the D3 brane case in [19], and the same argument applies here also. This strongly suggests that the gauge symmetries , should in fact be dual to the Betti multiplets discussed in §2.3.

Once we have identified the relevant abelian symmetries, we can consider acting with the and operations. We schematically write the action of the theory (which we will denote as ), separating the sector from the rest, as

 SU∼∫B3∧dB4+AI∧dAII+∫LR , (3.11)

where stands for the remaining terms. We can then consider a theory without the gauge fields , , constructed schematically as . By construction, this theory has exactly 2 global symmetries satisfying all the properties expected as dual to Betti multiplets. Following [22], we can introduce a background gauge field for one of them, which we can call . Then, as reviewed in §2.3, the -operation amounts to regarding this field as dynamical, while at the same time introducing a coupling to another background field as

 SSU→SSU[AI]+∫CI∧dAI . (3.12)

We can introduce yet another background gauge field for the second global symmetry and perform yet another -operation. However, this time we will choose to regard as the background gauge field on which to act with the -generator. This results in

 SSU[AI]+∫CI∧dAI→SSU[AI,AII]+∫CI∧dAI+∫CII∧d(CI−AII) . (3.13)

Integrating by parts yields

 SSU[AI,AII]+∫CI∧d(CII+AI)−∫CII∧dAII . (3.14)

Since only appears linearly, its functional integral gives rise to a delta functional setting , which leads to an action of the precise form (3.11). We have therefore been able to establish a connection between a theory where the gauge group is , and whose action is , with the original theory, whose action is given by , via repeated action with the -operation.

More generally, the whole of will act on the boundary conditions for the bulk gauge fields, leading in general to an infinite orbit of CFTs for each gauge symmetry in . This is a rich structure that deserves considerable further investigation. In this paper, however, we will content ourselves to study the particular choice of boundary conditions described by the theory. Since the dual to the operation is the exchange of the boundary conditions, we expect the gravity dual to the theory to still be , but with an appropriate choice of boundary conditions. In turn, these boundary conditions allow for the existence of the electrically charged M5 branes which we used to identify the symmetries. These M5 branes would not be allowed in the quantization , which in turn would be dual to a CFT where the corresponding factors would remain gauged. In agreement, the dual operators which we will propose below would not be gauge-invariant in that case.

Let us now consider the effect of the gauge group on the construction of the moduli space. The diagonalization of the auxiliary fields in the equations defining the moduli space (3.2) relies on the non-abelian part of the gauge symmetry, and therefore it applies even if we consider ungauging some of the diagonal factors. More crucially, in order to obtain the correct four-fold moduli space we needed the piece (3.3) of the CS action so that, upon dualizing the field, the dual scalar is gauge-fixed via gauge transformations of . Thus provided we leave and gauged, with the same CS action, all of this discussion is unaffected if we ungauge the remaining , . Correspondingly, we will still have the 8 gauge-invariants (3.8), which will give rise to the same 9 equations defining as a non-complete intersection as “mesonic” moduli space. The remarks on the non-abelian chiral ring elements spanned by (3.10) are also unchanged. However, with only a gauge symmetry we also have additional chiral primary operators, charged under the now global , . Indeed, we have the following “baryonic” type operators:

 BAI1...IN = 1N!ϵi1⋯iNϵj1⋯jN(AI1)j1i1⋯(AIN)jNiN , BBi = 1N!ϵi1⋯iNϵj1⋯jN(Bi)j1i1⋯(Bi)jNiNei(−1)i−1Nτ , BCi = 1N!ϵi1⋯iNϵj1⋯jN(Ci)j1i1⋯(Ci)jNiNei(−1)i−1Nτ . (3.15)

In particular, for the 6 fields in the quiver there is a canonical set of 6 baryonic operators given by determinants of these fields, dressed by appropriate powers of the disorder operators to obtain gauge-invariants under . These operators are in 1-1 correspondence with the toric divisors in the geometry. This is precisely the desired mapping between baryonic operators in the field theory and M5 branes wrapping such toric submanifolds, with one M5 brane state for each divisor. Indeed, the charges of these operators under the two baryonic s are

 BAI1..IN