Other Squashing Deformation and N=3 Superconformal Chern-Simons Gauge Theory

We consider one of the well-known solutions in eleven-dimensional supergravity where the seven-dimensional Einstein space is given by a SO(3)-bundle over the CP^2. By reexaming the AdS_4 supergravity scalar potential, the holographic renormalization group flow from N=(0,1) SU(3) x SU(2)-invariant UV fixed point to N=(3,0) SU(3) x SU(2)-invariant IR fixed point is reinterpreted. A dual operator in three-dimensional superconformal Chern-Simons matter theories corresponding to this RG flow is described.


Introduction
The N = 6 superconformal Chern-Simons matter theory with gauge group U(N) × U(N) at level k and with two hypermultiplets in the bifundamental representations is found in [1]. This gauge theory is described as the low energy limit of N M2-branes probing C 4 /Z k singularity. At large N-limit, this theory is dual to the eleven-dimensional M-theory on AdS 4 × S 7 /Z k where the seven-sphere metric is realized as an S 1 -fibration over CP 3 [2]. One of the main observations in [1] is to look at the special case of N = 3 superconformal Chern-Simons matter theory with above particular gauge group, matter contents, and particular choice of Chern-Simons levels of two gauge groups. Then naive SU(2) flavor symmetry appearing in the hypermultiplets is enhanced to SU(2) × SU (2) symmetry that occurs in the whole action of the theory and there exists a SU(2) R symmetry coming from the original N = 3 superconformal symmetry. It turns out the full theory has N = 6 superconformal symmetry and the full scalar potential is invariant under SU(4) R coming from this enhanced N = 6 superconformal symmetry.
The simplest spontaneous compactification of the eleven-dimensional supergravity is the Freund-Rubin [3] compactification to a product of AdS 4 spacetime and an arbitrary compact seven-dimensional Einstein manifold X 7 of positive scalar curvature. The standard Einstein metric of the round seven-sphere S 7 yields a vacuum with SO(8) gauge symmetry and N = 8 supersymmetry. There exists a second squashed Einstein metric [4,5] yielding a vacuum with SO(5) × SU(2) gauge symmetry and N = (1, 0) supersymmetry [6,7]. As suggested in [8,9], in [10], it was shown that the well-known spontaneous (super)symmetry breaking deformation from round S 7 to squashed one is mapped to a renormalization group(RG) flow from N = (1, 0) SO(5)×SU(2)-invariant fixed point in the UV to N = 8 SO(8)-invariant fixed point in the IR. In particular, the squashing deformation corresponds to an irrelevant operator at the UV superconformal fixed point and a relevant operator at the IR (super)conformal fixed point respectively. Moreover the RG flow is described geometrically by a static domain wall which interpolates the two asymptotically AdS 4 spacetimes with round and squashed S 7 's. For the different type of compactifications where the internal space has nonzero four-form field strength, see also [11,12]. One could ask [13] what happens when we perform Z k -quotient [1] along the above whole RG flow [10]? Starting from the general, one parameter-family, metric for CP 3 inside of sevensphere and its seven-dimensional uplift metric on an S 1 -bundle over this CP 3 , the full elevendimensional metric with appropriate warp factors was constructed. By analyzing the AdS 4 scalar potential, the holographic supersymmetric(or nonsupersymmetric) RG flow from N = (1, 0) SO(5) × U(1)-invariant UV fixed point to N = (6, 0) SU(4) R × U(1)-invariant IR fixed point was described in [13]. Each symmetry group is the subset of previous ones respectively. That is, SO(5) × U(1) is contained in SO(5) × SU(2) and SU(4) R × U(1) is contained in SO (8). The squashing deformation corresponds to the singlet of 20 ′ of SU(4) R and it is given by the quartic term for the matter fields transforming as fundamental representation under the SU(4) R . The dual Chern-Simons matter theory at the N = (1, 0) SO(5) × U(1)-invariant UV fixed point is constructed in [14]. Now it is natural to ask that are there any other examples where some squashing deformation in X 7 might provide a similar RG flow and one can think of some dual operator in three-dimensional boundary conformal field theory? Yes, the manifold X 7 = N 0,1,0 I has been studied originally by Castellani and Romans [15] who identified this manifold as a particular coset manifold which is specified by three integers. We consider the particular case where p = 0, q = 1 and r = 0 in N p,q,r I manifold. There exists N = (3, 0) supersymmetry with SU(3) × SU(2) gauge symmetry or N = (1, 0) supersymmetry with SU(3) × U(1) gauge symmetry(See also [16]). Moreover, Page and Pope [17] have completed the coset manifold construction by showing existence of another family of Einstein manifold, N 0,1,0 II , which can be obtained from geometric squashing of the N 0,1,0 I , retains the same gauge group SU(3) ×SU(2) but instead preserves N = (0, 1) supersymmetry. As in the case of seven-sphere S 7 , the scalar field corresponding to the squashing deformation acquires a nonzero vacuum expectation value leading to (super)-Higgs mechanism. With left-orientation the squashing interpolates between a N = 3 supersymmetric vacuum and another with N = 0 supersymmetry. With right-orientation it interpolates between a nonsupersymmetric vacuum and a supersymmetry restored one with N = 1 supersymmetry [18].
On the other hand, in [19], the corresponding N = 3 dual gauge theory has gauge group SU(N) × SU(N) with "three" hypermultiplets transforming as a "triplet" under the SU(3) flavor symmetry which is nothing but one of the global symmetries for N 0,1,0 I manifold 1 . In terms of N = 2 superfields, these hypermultiplets can be reorganized as two sets of chiral superfields. For the color representation, one of these transforms as (N, N) and the other transforms as (N, N). Furthermore, these two superfields transform as a doublet of SU(2) R which is the SU(2) factor in the remaining global symmetry of N 0,1,0 I manifold. As we mentioned before, SU(2) R corresponds to the N = 3 superconformal symmetry. After integrating out two adjoint fields of the theory, the effective quartic superpotential can be obtained. The coefficient is determined by the N = 3 supersymmetry but breaks N = 4 supersymmetry. For the general discussion on N = 3 superconformal Chern-Simons matter theory, see [20]. The complete N = 3 Kaluza-Klein spectrum is found in [21] and its OSp(3|4) multiplet structure is further explained in [22].
Later, Billo, Fabbri, Fre, Merlatti and Zaffaroni [23](See also [24]) have constructed an N = 3 long massive spin 3/2 multiplet with conformal dimension 3 from the massless N = 3 graviton multiplet. These two are connected to "shadow" relation: fields of different type, spin and mass are linked by a relation which determines the mass of the one as a function of the other.
In this paper, we will be studying the known example of Kaluza-Klein supergravity vacua and reinterpret it in terms of three-dimensional (super)conformal field theories and associated RG flows. We will be exploring the Freund-Rubin type spontaneous compactification on AdS 4 × X 7 . For M2-branes on an eight-dimensional manifold, the near-horizon geometry X 7 is expected to change as the M2-branes are placed at or away a conical singularity of the manifold [25,26]. More specifically, we will consider X 7 being 3-Sasaki holonomy manifold, describing near-horizon geometry of M2-branes at relevant conical singularities.
When the work of [18] was completed at that time, it was not possible to analyze the gauge theory description because the Kaluza-Klein spectrum of eleven-dimensional supergravity was not complete. Later, in [23], they have found more mass spectrum in the eleven-dimensional supergravity side that includes the harmonics of the Lichnerowicz scalar with conformal dimension 4. Then one can identify the corresponding fluctuation spectrum for the scalar fields around N = 3 fixed point.
The aim of this paper is to 1) recapitulate the effective scalar potential described in [18] with only breathing mode and squashing mode, and 2) analyze more both the mass spectrum in the eleven-dimensional supergravity and the corresponding Chern-Simons gauge theory operator which gives rise to the squashing deformation, by analyzing the results of [23].
In section 2, we describe the seven-dimensional Einstein space(N 0,1,0 I ) and its squashed version(N 0,1,0 II ) compactification vacua in eleven-dimensional supergravity. The effective fourdimensional scalar potential looks similar to the one for seven-sphere and the two critical points have nonzero scalar fields. However, the ratio of the squashing parameter at these two critical values(which is equal to 1/5) is the same as the one in seven-sphere case.
In section 3, the squashing deformation of each vacua is described by an irrelevant operator at the N = (3, 0) conformal fixed point and a relevant operator at the N = (0, 1) conformal fixed points. The RG flow is described in AdS 4 supergravity by a static domain wall interpolating between these two vacua. We identify the corresponding operator in the boundary conformal field theory in three dimensions by looking at the observations of [23] 2 .

Two seven-dimensional Einstein spaces
A generic eleven-dimensional metric interpolating between two seven-dimensional Einstein spaces with an arbitrary four-dimensional spacetime metric maybe written as where the three real left-invariant one-forms satisfy the SU(2) algebra dσ i = − 1 2 ǫ ijk σ j ∧ σ k and those on the manifold SO(3) which is necessary for the regularity of the metric have The coordinate µ and SU(2) one-forms σ i are the same as CP 2 metric and corresponding isometry is characterized by SU (3). Then the seven-dimensional N 0,1,0 space is a nontrivial SO(3) bundle over CP 2 and the isometry group for this space is SU(3) × SU(2) [17]. The scalar fields u(x) for the breathing mode and v(x) for squashing mode depend on the four-dimensional spacetime. Moreover, the squashing is parametrized by [34] The parameter R measures the overall radius of curvature. The gauge fields are given by A 1 = cos µ σ 1 , A 2 = cos µ σ 2 and A 3 = 1 2 (1 + cos 2 µ)σ 3 . Spontaneous compactification of M-theory to AdS 4 × X 7 is obtained from near-horizon geometry of N coincident M2-branes and the nonvanishing flux of four-form field strength of the Freund-Rubin [3] is given by Here the Page charge [34,32] p . Then the eleven-dimensional Einstein equation 2 Recently, the N = 3 superconformal Chern-Simons quiver theories are constructed in [27] but these theories do not contain the seven-dimensional Einstein manifold we are considering here. See also other relevant paper [28]. For the earlier studies on N = 3 superconformal Chern-Simons theories, there are also some works in [29,30,31]. 3 They are given by σ 1 = cos ψdθ + sin ψ sin θdφ, σ 2 = − sin ψdθ + cos ψ sin θdφ and σ 3 = dψ + cos θdφ and similarly SO(3) one-forms are given by Σ 1 = cos γdα + sin γ sin αdβ, Σ 2 = − sin γdα + cos γ sin αdβ, and Σ 3 = dγ + cos αdβ [32,33]. with (2.3) provides the following Ricci tensor components [34,13] On the other hand, the Ricci tensor components can be obtained from the following orthonormal basis which can be read off from the eleven-dimensional metric (2.1) The results for Ricci tensor in the basis of (2.5) are summarized by Substituting the last two relations in (2.6) into (2.4) implies the field equations for the breathing mode u(x) and the squashing mode v(x) as follows: Note that the only first two terms in u ;α ;α and v ;α ;α are different from the one for squashed S 7 space [34,13]. The vanishing of the right hand side of second equation of (2.7) implies that either v = v 1 = 1 7 ln 2 or v = v 2 = 1 7 ln 10. Furthermore, substituting the field equation 4 When the scalar fields u(x) and v(x) are constant, then R where (2.2) is used. Then λ 2 = 1 2 corresponds to the Einstein metric in [15] and λ 2 = 1 10 corresponds to the squashed Einstein metric [17].
for u(x) in (2.7) into the first equation of (2.6) together with (2.4) implies the following four-dimensional Ricci tensor Now the field equations (2.7) and (2.8) are equivalent to the Euler-Lagrange equations for the following effective Lagrangian with the scalar potential 5 (2.10) Note that the first two terms in (2.10) are different from the one for squashed S 7 space [34,10,13]. One analyzes two vacua of this scalar potential as follows: 3 4 Q 2 , v = v 2 = 1 7 ln 10, λ 2 = 1 10 , The two supergravity solutions are classically stable under the changes of the size and squashing parameter of seven-dimensional space [36]. The gives rise to a theory with no supersymmetry(N = 0) [17].
Moreover, the left-handed squashed seven-dimensional space N 0,1,0 II,L gives rise to a theory with no supersymmetry while the right-handed squashed seven-dimensional space N 0,1,0 II,R gives rise to a theory with N = 1 supersymmetry. That is, with the choice of left-handed orientation of N 0,1,0 L , one regards the λ 2 = 1 2 metric as giving the unbroken vacuum state with N = 3 supersymmetry which can be broken spontaneously to λ 2 = 1 10 metric yielding N = 0 supersymmetry. On the other hand, with the choice of opposite orientation of N 0,1,0 R the λ 2 = 1 10 metric provides a vacuum state with N = 1 supersymmetry which can be broken to the λ 2 = 1 2 metric with N = 0 supersymmetry. Therefore, for the squashing with left-handed orientation, the RG flow interpolates between the boundary conformal field theories with N = 3 and N = 0 supersymmetry while for the squashing with right-handed orientation, the RG flow interpolates between conformal field theories with N = 0 and N = 1 6 . . Following [34,10,13], it is more convenient to rewrite (2.9) in terms of the unrescaled M-theory metric g αβ = e −7u g αβ in (2.1):
in which the un-rescaled cosmological constant Λ 1 = e 7u 1 Λ 1 = 1 2 e 7u 1 V (u 1 , v 1 ) is given by Here r IR is related to N and Planck scale ℓ p as r IR = ℓ p 1 2 (32π 2 N) 1/6 . By rescaling the scalar fields from the kinetic terms in the Lagrangian (3.1) as one obtains the fluctuation spectrum for v-field around the N 0,1,0 I which takes a positive value: where the relation (3.2) is used.
Recall that in the compactification of AdS 4 × S 7 , the v-field represents the squashing of S 7 and hence ought to correspond to 300(that is the Young tableaux of SO(8) or the SO(8) Dynkin label is given by (0, 2, 0, 0)): the lowest mode of the transverse, traceless symmetric tensor representation. The branching rule of the representation 300 in terms of SO(7) Dynkin labels is given by [38,39]  According to the nice observations of [23], the harmonics of Lichnerowicz scalars with conformal dimension ∆ = 4 has been obtained. The harmonics are eigenfunctions of the Lichnerowicz operator with eigenvalue M 200 = 96m 2 in their notation. Recall that the representation (2, 0, 0) refers to SO(7) representation 27. With the help of [40], the value of M 200 can be obtained from M3 = −16, one gets M 200 = 96m 2 . Note that m 2 is mass-squared parameter of a given AdS 4 spacetime and the mass of a scalar field φ in [40] is defined as (∆ AdS + M 200 − 32m 2 )φ = 0. Then 96m 2 goes to 64m 2 by subtracting 32m 2 and by dividing out 16 further in order to compare with the usual normalization in AdS/CFT correspondence, one arrives that the correct result for the masssquared is 4m 2 which is the same normalization used in [18]. For example, see also the recent paper [41] for the normalization between the conformal dimension and the mass-squared term.
Therefore, the mass-squared for the representation 6 of SU(3) is given by which is exactly equal to (3.3). Note that the assignment of SU(2) R isospin J is related to the SU(3) assignment and the R-symmetry group is the maximal SU(2) subgroup of SU (3).
Under this embedding, the SU(3) representations decompose into as follows: Due to the skew-whipping, the theory will be either left-squashed N 0,1,0 II,L with N = 0 supersymmetry or right-squashed N 0,1,0 II,R with N = 1 supersymmetry. The isometry of the squashed seven-dimensional Einstein manifold is given by SU(3) × SU (2). In terms of the unrescaled M-theory metric, the Lagrangian (2.9) can be rewritten as where the scalar potential is given by and the un-rescaled cosmological constant Λ 2 = e 7u 2 Λ 2 = 1 2 e 7u 2 V (u 2 , v 2 ) is given by The mass spectrum of the v(x) field is calculated similarly From the mass formula for the SU(3) × SU(2) representation and the eigenvalues of the Lichnerowicz operator, one should obtain the mass-squared for the singlet as follows: M 2 (1,1) = − 20 9 m 2 2 , and this coincides with (3.4). The perturbation that corresponds to squashing around N 0,1,0 II has a scaling dimension either ∆ = 4/3 or 5/3 and hence corresponds to a relevant operator.
We gave a nonzero expectation value to a supergravity scalar in the 6 of SU(3). Using the AdS/CFT correspondence, one identifies this perturbation with a composite operator of N = 3 superconformal Chern-Simons matter theory with a mass term for the symmetric and traceless product between two 3's: λ AB d 3 xO AB where λ AB is in the 6 of SU (3). Note that the tensor product of these leads to 3( ) × 3( ) = 3( ) ⊕ 6( ). Then one can construct a 3(that is the Young tableaux ) representation by using the Clebsch-Gordan coefficient together with matter field C I : Γ AIJ C I C J where C I (I = 1, 2, 3) are three complex scalars(3 under the SU(3)) transforming as (N, N) with gauge group SU(N) × SU(N) in N = 3 superconformal Chern-Simons gauge theory [19]. The perturbation is given by The singlet of this operator O AB which is 6 of SU(3) corresponds to the supergravity field v(x)(or v(x)) and the conformal dimensions are given by ∆ U V = 4 3 (or 5 3 ) and ∆ IR = 4 respectively as we computed before. The other five states with J = 2 among 6 of SU(3) are non-diagonal and correspond to deformations of the seven-dimensional metric [23]. Since the Lichnerowicz operator provides nine-dimensional space, the remaining three states correspond to the eigenvalue M 200 = 0. These are organized in a triplet(J = 1) of SU(2) R and the massless scalars belong to the additional massless vector multiplet. So far we considered only FR compactification where there are no internal components for the four-form field strength. In [23], they further described the deformation from turning on an internal three-form.

Conclusions and outlook
We have constructed the full eleven-dimensional metric given by (2.1) and obtained the scalar potential in (2.10) by using the Freund-Rubin ansatz (