Higgs-like (pseudo)scalars in AdS4, marginal and irrelevant deformations in CFT3, and instantons on S 3

Employing a 4-form ansatz of 11-dimensional supergravity over a non-dynamical background and setting the internal space as an Hopf fibration on , we obtain a consistent truncation. The (pseudo)scalars, in the resulting scalar equations in Euclidean AdS space, may be considered to arise from (anti)M-branes wrapping around the internal directions in the (Wick-rotated) skew-whiffed M2-brane background (as the resulting theory is for anti-M2-branes), thus realizing the modes after swapping the three fundamental representations , , and of . Taking the backreaction on the external and internal spaces, we obtain the massless and massive modes, corresponding to exactly marginal and marginally irrelevant deformations on the boundary CFT , respectively. Subsequently, we obtain a closed solution for the bulk equation and compute its correction with respect to the background action. Next, considering the Higgs-like (breathing) mode , having all supersymmetries as well as parity and scale-invariance broken, solving the associated bulk equation with mathematical methods, specifically the Adomian decomposition method, and analyzing the behavior near the boundary of the solutions, we realize the boundary duals in the -singlet sectors of the ABJM model. Then, introducing the new dual deformation operators made of bi-fundamental scalars, fermions, and gauge fields, we obtain the -invariant solutions as small instantons on a three-sphere with the radius at infinity, which correspond to collapsing bulk bubbles leading to big-crunch singularities.


Introduction
Euclidean solutions with finite actions, known as Instantons, as non-perturbative phenomena play important roles in physics from quantum corrections to classical system behavior to early universe cosmology. 1In a series of studies-see [1], [2], [3], [4], [5] as recent ones-we have presented a number of such solutions in the context of AdS 4 /CFT 3 correspondence, the best model of which is ABJM [6].
In fact, the ABJM action describes the world-volume of N intersecting M2-branes on an Z k orbifold of C 4 (four complex coordinates), with the orbifold acting as X A → e 2πi k X A , A = 1, 2, 3, 4. In the 't Hooft large N limit and fixed λ = N/k, the 11-dimensional (11D) supergravity (SUGRA) over AdS 4 × S 7 /Z k is valid when N ≫ k 5 , and by the orbifold, the subgroup SU(4) × U(1) ≡ H of the original SO(8) ≡ G remains.The 3D boundary theory is a U(N) k × U(N) −k Chern-Simon (CS) gauge theory with N = 6 supersymmetry (SUSY) and matter fields (scalars Y A and fermions ψ A ) in bi-fundamental representations (reps) (4 1 and 4−1 ) of H.
Here, by keeping the ABJM background geometry unchanged and taking a 4-form ansatz of the 11D SUGRA, composed of the ABJM internal CP 3 ⋉ S 1 /Z k space elements and scalars in the external 4D Euclidean anti-de Sitter (EAdS 4 ) space, associated with probe (anti)M-branes wrapped around mixed directions in (M2-branes)anti-M2-branes background resulting in anti-M2-branes theory, we will have a consistent truncation in that just H-singlet fields remain in the truncated theory and all dependencies on the internal 7D space are omitted in resulting equations. 2n the other hand, instantons as topological objects should not backreact on the background geometry.To this end, we solve the truncated equations in EAdS 4 together with the equations resulting from zeroing the energy-momentum (EM) tensors of the Einstein's equations that result in equations for massless and massive bulk (pseudo)scalars, which in turn correspond to exact and irrelevant marginal deformation of the dual boundary theory.In addition, we consider a Higgs-like mode (m 2 = 18), already known as breathing mode; see [8], [11], [12] and [10].To get solutions for the bulk scalar equations, we employ the usual mathematical methods and especially the Adomian Decomposition Method (ADM) [13] to solve the Nonlinear Partial Differential Equations (NPDEs).
Next, after analyzing the bulk solutions near the boundary and dual symmetries, we propose the corresponding dual operators to deform the boundary action with and find solutions.In fact, because the bulk setups and solutions break all SUSYs, N = 8 → 0, parity-and scaleinvariance, to realize the boundary duals, we swap the three fundamental reps 8 s , 8 c and 8 v of SO (8).With such a swapping, we could realize the H-singlet scalars and pseudoscalars in the mass spectrum of the 11D SUGRA on the background geometry after the branching G → H, corresponding to H-singlet boundary operators.Meanwhile, as the scale symmetry is violated because of the mass term in the equations and their extreme nonlinearity, the solutions should preserve the SO(4) symmetry of the original isometry SO(4, 1) of EAdS 4 .Keeping a singlet sector of the boundary ABJM action, 3 with only one scalar, one fermion and U(1) × U(1) part of the gauge group, we find such an SO(4)-invariant solutions with finite actions, which are always small instantons triggering instabilities on a three-sphere with radius r at infinity (S 3 ∞ ), and describe big crunch singularities in the bulk.
This article is organized as follows: In section 2, we present the 11D SUGRA background, including the 4-form ansatz, and the equations for (pseudo)scalars in EAdS 4 ; Emphasizing that in the anti-M2-branes background, a (pseudo)scalar becomes Higgs-like providing spontaneous symmetry breaking and making the main equation homogeneous.In subsection 2.1, we consider backreaction; that is after computing the EM tensors of the Einstein's equations in Appendix A, from zeroing them and solving the resulting scalar equations with the main one in the bulk, we get solvable PDEs for the massless and massive (pseudo)scalars from taking the backreaction on the external and internal spaces, respectively.Next, in subsection 2.2, we present an exact solution for the equations in the latter subsection and compute its corrections to the background action.In section 3, we employ known methods of differential equations to solve the main Higgs-like NPDE and get solutions near the boundary.In particular, in subsection 3.1, we use the ADM (with details in Appendix B) to obtain solutions appropriate for near the boundary analyzes of the Higgs-like mode m 2 = 18, up to the third order of the perturbative series expansion.In section 4, we first discuss dual symmetries from the bulk setups, equations and solutions and next, deal briefly with the spectrum of 11D SUGRA over AdS 4 ×S 7 /Z k and check whether we can find the desired H-singlet scalars and pseudoscalars among various generations after swapping the fundamental reps of SO (8) for gravitino and the branching of G → H; Then, we present basic elements of AdS 4 /CFT 3 correspondence for (pseudo)scalars needed for our boundary analyzes.In section 5, we look for dual solutions in ABJM-like 3D field theories; and in this way, in subsections 5.1 and 5.2, we consider marginal and irrelevant deformations with new H-singlet ∆ + = 3, 6 operators, corresponding to the massless (when taking the backreaction) and massive bulk states, respectively, and find SO(4)-invariant solutions with finite actions as instantons.In addition, from the boundary solutions, we confirm the stateoperator correspondence, match the bulk-boundary parameters and determine an unknown scalar function in a bulk solution from the correspondence.Meanwhile, we remind that with a marginal triple-trace deformation of a dimension-one operator composed of bi-fundamental scalars, we could build the tri-critical O(N) model and find Fubini-like instantons.Also, we will confirm the Bose-Fermi (BF) duality between a deformation with the latter ∆ + = 3 operator (in fact the massless Regular Boson (RB) model) and a deformation with a ∆ + = 6 operator composed of bi-fundamental fermions (in fact the massless Critical Fermion (CF) model) at least at the level of solutions and correspondence, in subsection 5.2.Eventually, in section 6, we present a summary along with comments on solutions, physical interpretations, connections with other studies and related issues.

From 11D Supergravity to 4D Gravity Equations
We start with the 4-form ansatz4 for 11D SUGRA over AdS 4 × S7 /Z k when the internal space is considered as a U(1) bundle on CP 3 , where 3 = NE 4 is for the ABJM [6] background with N = (3/8)R 3 units of flux quanta on the internal space, R = 2R AdS is the AdS curvature radius, E 4 is the unit-volume form on AdS 4 , J is the Kähler form on CP 3 , e 7 is the seventh vielbein5 (of the internal space) and f i 's with i = 1, 2, 3 are scalar functions in bulk coordinates.
Taking the ansatz (2.1), from the Bianchi identity and Euclidean 11D equation we get where C i 's are real constants, N f 1 ≡ f1 and note that the plus and minus signs on the last term of the RHS equation indicate considering the Wick-rotated (WR) and skew-whiffed (SW) backgrounds respectively, and that the ABJM background realizes with C 3 = 1.In addition, from the equation (2.2), making use of (2.3), we get 6 where * 4 d ( * 4 df 3 ) = 4 is the EAdS 4 Laplacian, and we use the following conventions: Next, from (2.3) and (2.4), we write 8 where (2.7) To make the equation (2.6) homogeneous (that is F = 0), besides C 2 = 0, we have to set and so where m2 is indeed the squared mass of f 3 in (2.4).To have physically permissible (nonimaginary) masses in this case, we have just to consider the SW version with C 3 ≥ 1/3; and as a result, the SW ABJM background realizes with C 3 = 1 and then m 2 R 2 AdS = +4 (C 2 = 2/ √ 3).In addition, as noticed in [5], ± (C 2 /2) = ± − m2 /λ are in fact homogenous vacua and so, the (pseudo)scalar is Higgs-like and the LHS relation in (2.3) imposes spontaneous symmetry breaking, where f acts as fluctuation around the homogeneous vacua.

Taking Backreaction and Resulting Equations
To take backreaction, we should first compute the EM tensors of the corresponding Einstein's equations, the details of which are in Appendix A. In fact, since we are looking for instantons that, as topological objects, should not backreact on the background geometry, we solve the main bulk equations with the equations in Appendix A, resulting from zeroing the EM tensors.
In this way, we first see that the equation (A.7) is solved with (2.6) with which means taking the backreaction of the external AdS 4 space on the background geometry gives the massless m 2 R 2 AdS = 0 bulk (pseudo)scalar replying to the boundary exactly marginal operators. 9s the same way, noting that the equation (A.9) is the same as the main one (2.6), from solving the equations (A.8) as well as (A.7) and (A.8) with (2.6), that is taking the backreaction of the internal (indeed CP 3 ) and whole 11D space, we get respectively, with m 2 R 2 AdS = 1/2, 2/9 corresponding to the marginally irrelevant ∆ ± = 3/2 ± √ 11/2, 3/2 ± (89/9)/2 boundary operators, which we encountered the former recently in [4].

A Solution For the Case with Backreaction
One may solve the equations (2.11) and (2.12) using the usual mathematical methods, such as separation in variable.However, a well-known closed solution for the equations-leaving out the inhomogeneous terms that do not contribute to the dynamics10 -reads [18], [19] , and we use the EAdS 4 metric noting u = (x, y, z), in upper-half Poincaré coordinates and so On the other hand, because the bulk solutions including the backreaction correspond to variants of marginal operators, for simplicity we consider the instanton solution for (2.10) and compute its correction to the background action.To this end, as the background geometry does not change, we use the right parts of the bosonic action of 11D SUGRA in Euclidean space as where κ 2 11 = 9π G 11 = 1 4π (2πl p ) 9 , and κ 11 , G 11 and l p are the 11D gravitational constant, Newton's constant and Plank length, respectively.
Next, from the ansatz (2.1), using (2.5), we write its 11D dual 7-form as (2.17) and By placing the latter relations in (2.16), using (2.3) (noting that with ) where the first term on the RHS is the contribution of the ABJM background realized with C 3 = 1, as one may see from the second term on the RHS relation in (2.3), and the last (surface) term, as a total derivative, does not contribute to the equations and we discard it.
Then, to compute the action (2.19) based on the solution (2.13) with ∆ + = 3, we use and the 3D spherical coordinates, setting | u − u 0 | = r.As a result, the finite contribution of the action, after integrating on the external space coordinates, in the unit 7D internal volume, reads Scorr.
where ĉ ≃ 0.000016 and č ≃ 0.0033, and because of singularities, we have included ǫ > 0 as a cutoff parameter to evade the infinity of integrals with respect to (wrt) u; see [20].Meantime, we note that for finite k and R, (2.21) is a small contribution.

Solutions For the Higgs-Like Scalar Equation
The Higgs-like (pseudo)scalar equation of (2.6), with (2.8) and (2.15), reads 11 For its linear part, using the spherical coordinates with r = | u|, discarding the angular parts, and separation of variables, f 0 (u, r) = f (r)g(u), we can write with combinations of Hyperbolic and Bessel (or with k = iκ, Trigonometric and Modified Bessel) functions as solutions for f (r) and g(u), respectively. 12Then, one may use the leading 11 From now on, we use f 2 ≡ f and the plus sign for the f 2 term in the equations. 12An interesting solution for the r part is f (r) ∼ e −r /r, which might be considered as constrained instantons; see for instance [21] and also [4], where we discussed a similar solution in a 3D boson model.Another interesting solution is where C1 and C2 are constants and ℓ(ℓ + 1) = κ 2 .
order (LO) solutions to get the higher-order solutions of the full NPDE.The resulting solutions always reproduce the right behavior of (pseudo)scalars near the boundary as On the other hand, one may employ an ansatz like which turns (3.1) into the following NODE where we define As a result, the appropriate part of a perturbative solution for (3.6), up to the first or nextto-leading order (NLO), reads 13 where C l 's are real constants.
As the same way, by ξ = r/u (the so-called self-similar reduction method via the scaleinvariance of the variables; see for instance [22]), the equation (3.1) turns into A solution for the linear part of the latter equation is in terms of Legendre functions, and from that, one may build perturbative series solutions up to higher-orders; for such a solution, see [4].Alternatively, we can also use which turns the equation (3.10) into the following first-order Riccati equation (3.12) 13 It is recalled that we generally use For massive modes, a common series solution for the latter equation, keeping the normalizable term appropriate for the corresponding boundary analyzes of AdS 4 /CFT 3 , reads from which one may build higher-order solutions.For example, for the mode m 2 = 18 that we consider, a series expansion around u = 0, up to NLO, reads with the real constants Č∆ − and Ĉ∆ + -when doing boundary analyzes, we return to this solution as well.Meanwhile, from near the boundary behavior of the closed solution of (2.13), we can read f (r) = C6 /r 12 to rewrite the series solutions clearly.
On the other hand, we can use (B.8) with (B.9) and near the boundary behavior of the closed solution of (B.7), as the initial data, which might also be read from the LHS relation in (B.1), in the ADM, to get approximate solutions.As a result, we arrive at a series solution about u = 0, up to the first iteration of ADM or NLO of the expansion, as where H ∆ + (r, a 0 , b 0 , m) is a polynomial of its arguments; and in particular, for the term corresponding to the bulk mode m 2 = 18, it becomes 4 Dual Symmetries, Mass Spectrum and Correspondence First, we remind that the truncation here is consistent, considering that our ansatz (2.1) is H-singlet, given that e 7 , J and the (pseudo)scalars in resulting equations respect the same symmetry.Second, the setups here are as if we add ℓ probe (anti)M-branes to the (WR)SW M2-branes background and so, the resultant theory is for anti-M2-branes with the quiver gauge group of SU(N + ℓ) k × SU(N) −k .Indeed the (anti)M-branes wrap around mixed internal and external directions and so break all SUSY's and parity 14 , and that to realize the latter we focus on U(1) ⊂ U(ℓ) part of the gauge group (in the large k limit) and keep G as a spectatora so-called novel Higgs mechanism; see for instance [26]. 15Third, the bulk settings break the inversion (and so, the special conformal transformation K µ ) symmetry and scale-invariance (denoted by the dilation operator D) because of the mass and nonlinear terms in the bulk action and translational-invariance (denoted by the translation operator P µ ) because of nonconstant solutions. 16As a result, the conformal symmetry SO(4, 1) (as the isometry of EAdS 4 ) breaks into SO(4), which in turn includes six generators consisting of three Lorentz transformations (denoted by the operator L µν ) and R µ ≈ (K µ + a 2 P µ ) 17 corresponding to rotations on S 3 , where a is the scale parameter.The four generators of the broken symmetries (translations and scale transformations)-and so the four free parameters a (or b 0 ) and u 0 -move the SO(4)-symmetric (SO(3, 1) in Lorentzian signature) bubble around in the 4D bulk.
On the other hand, the mass spectrum of 11D SUGRA over AdS 4 × S 7 /Z k18 includes three generations of scalars (0 + 1 , 0 + 2 , 0 + 3 ) and two generations of pseudoscalars (0 − 1 , 0 − 2 ).In fact, the massless multiplet (n = 0) includes a graviton (1), a gravitino (8 s ), 28 spin-1 fields ( 28 with n = 2 of G, again without any H-singlet under the branching, except for the last one 1(0, 0, 0, 0) → 1 0 [0, 0, 0].However, because of the triality of G20 , one can exchange its three inequivalent reps 8 v , 8 s , 8 c ; In fact, to find the desired singlet modes and realize SUSY breaking in the boundary theory, we swap the three reps 21 .Therefore, with the swapping 8 s ↔ 8 c and 8 v fixed, which means exchanging spinors(supercharges) with fermions and keeping scalars unchanged, the massless and massive pseudoscalar reps change accordingly without any H-singlet under the branching of the resulting reps, while the scalar reps do not change.As the same way, after the swapping 8 s ↔ 8 v and 8 c fixed, which means exchanging spinors with scalars and keeping fermions unchanged, the resulting reps of both modes as pseudoscalar do not include any H-singlet under the branching.However, we have 1386 s and 30940 s , 23400 s from the massless and massive scalar modes, respectively, while the rep 1 of 0 + 2 with n = 2 of G remains the same as before, with the latter swapping.For the latter reps, the branching G → H reads 30940 s → 1 0 ⊕ 20 0 ⊕ 105 0 ⊕ 336 0 ⊕ 8 25 0 ⊕ 1716 0 ⊕ 3185 0 ⊕ ... , where we have only written U(1)-neutral reps, and that the corresponding reps for 1386 s are the same as the first four terms of the reps above for 30940 s under the branching.As a result, we see that after exchanging s ↔ v, there is the desired H-singlet rep (1 0 ) for both massless and massive (pseudo)scalars we consider here.On the other hand, a bulk (pseudo)scalar with near the boundary behaviour of (3.4) could be quantized with either the Neumann or alternate (δβ = 0) boundary condition for the masses in the range of −9/4 ≤ m 2 ≤ −5/4 or the Dirichlet or standard (δα = 0) boundary condition that can in turn be applied to any mass (see for instance [41] and [42]), while the regularity (that ∆ + is real) and stability require that the mass is above the Breitenlohner-Freedman (BF) bound m 2 ≥ m 2 BF = −9/4 [43], [44].As a result, for the massless and massive modes, only mode β is normalizable; and α and β have holographic expositions as source and vacuum expectation value of the one-point function of the operator ∆ + , and vice versa for the operator ∆ − .Then, we write the Euclidean AdS/CFT dictionary as where ) is the generating functional of the connected correlator of the operator O ∆ + (O ∆ − ) on the usual (dual) boundary CFT 3 with ∆ + (∆ − ) quantization.

Dual Solutions in Boundary 3D Field Theories
The bulk setups with the symmetries discussed in the previous section 4, including parity breaking, are dual to the boundary CS O(N) or U(N) interacting vector models. 22However, we usually consider elements of ABJM's model with, depending on the case, only one scalar (say Y = ϕ = h(r) I N , with h(r) as the scalar profile) or fermion (say ψ) 23 resulting in zero scalar and fermion potentials, and a deformation as where the CS Lagrangian reads which is attributed to the remaining U(1) part of the original quiver gauge group discussed in section 4 24 , ∆ , whose integral is W in (4.2), stands for (with p marking) deformations we make with various H-singlet operators.

Marginal Deformations and Solutions For the Massless State
For the bulk solutions in subsections 2.1 and 2.2, arising from taking the backreaction, which correspond to (exactly and irrelevant 25 ) marginal operators, besides the ∆ + = 3 operators of O 22 It is noticeable that according to [45], nonlinear Higher-Spin gauge theories violating parity in AdS 4 correspond to nonlinear interacting 3D boundary CFTs. 23The singlet (pseudo)scalar or fermion we consider could be taken from decomposing the eight (pseudo)scalars or fermions as X I → (Φ n , Φ, Φ), with Φ representing either ψ or Y , I, J... = (1, ...6, 7, 8) = (n, 7, 8) and Φ = Φ 7 + i Φ 8 , Φ † = Φ, transforming in the rep (6 0 , 1 2 , 1 −2 ) under SO(8) → SU (4) R × U (1) b . 24We may also take the U (1) × U (1) part of the gauge group, with the gauge fields A ± i ≡ (A i ± Âi ), noting that the fundamental fields of ABJM are neutral wrt A + i (diagonal U (1)) and A − i acts as baryonic symmetry; and since our (pseudo)scalars are neutral, we set A − i = 0.As a result, we will also examine the sum of L CS (for A i ) and LCS (for Âi ) instead of L + CS in the boundary analyzes. 25See footnote 28.considered in [16], [17], [46], [2], [3], [4] and [5], here we include two new ones: Next, we consider a deformation as where the λ's are coupling constants, and set α = 1 for now.Then, if we take both CS terms L CS + LCS instead of L + CS in (5.1), after some mathematical manipulations on the resultant scalar φ = ϕ † , fermion ψ and gauge A + k field equations, we get where Y = Y † and A − i = 0 are also set.After that, a closed solution for (5.5) reads where g 6 ≡ −λ 6 , while by employing the ansatz where A + (r) is a scalar function on the boundary, a solution for (5.6) reads (5.9) where γ = (σ 2 , σ 1 , σ 3 ) are the Euclidean gamma matrices and χ with χ † χ = 1 is a constant dimensionless spinor 26 .Finally, from computing the corresponding boundary action based on the solutions (5.7), (5.9) and (5.10), setting the couplings equal to 1 and ã = a = a † for simplicity, we get S modi. (3) π 2 , (5.12) 26 See [47] for a similar ansatz.
27 which is finite, indicating an instanton with size a ≥ 0 at the origin ( u 0 = 0) of a three-sphere with radius r at infinity (S 3 ∞ ).As a result and a basic test of the correspondence, wrt (4.2), we have with a 1 , a 2 , ... as boundary constants, which is compatible with near the boundary behavior of (3.13) with ∆ + = 3, wrt (3.4), in the limit of a → 0, r → ∞.Meantime, comparing with the bulk closed solution of (2.13), this boundary solution may be considered as an instanton sitting at the conformal point of u = a. 28ote also that with λ6 = λ6 = 0 in (5.4) and including a mass-deformation term (m 2 b tr(ϕ φ)), 29 we have in general the RB model (look also at [4]), whose φ equation reads Solutions for its free massive equation are available in terms of (modified) Bessel functions, with an explicit one as which satisfies the condition h c (r → ∞) → 0, resulting in a finite action.Solutions for the interaction equation could be obtained in the context of constrained instantons; see [55], [56], [21] and [57].In fact, making use of (5.15) as the initial data, one may employ perturbative methods and get solutions with a simple structure like h ∼ 1/r and so, we have the singleoperator correspondence O (a) 3 α ∼ 1/r 6 with the typical near the boundary solution of (3.13) for ∆ + = 3.
In particular, if we use just the operator O (e) 3 to deform the action of (5.1) with, discarding its fermion kinetic term, the equations for φ and ) respectively.Next, with ϕ = φ, from the last two equations, we can write while for the gauge field, we may use the ansatz (5.8) with A + (r) ∼ 1/r 2 and so, with a 3 = 0, the basic correspondence O (e) 3 α ∼ 1/r 6 is realized, with near the boundary solution of (3.13) with ∆ + = 3, wrt (3.4).On the other hand, if ϕ = φ, which is allowed due to being in Euclidean space, and explicitly with from the equations (5.16) and (5.17), we get The latter solution is reminiscent of the duality (A k = η kj ∂ j h/h) between the instanton solution of the pure SU(2) Yang-Mills theory [58], with η ij as 't Hooft symbols [59], and the SO(4)-invariant solution of the so-called ϕ 4 model as 30 ( As a result, with the latter solutions, we have the same correspondence as (5.13) for O (e) 3 .
It is also interesting to check the BF duality (or 3D Bosonization)-see for instance [67] and [68]-from our setups attributed to RB and CF models31 at the level of the solutions.Indeed, under the BF duality, the coupling of g6 tr(ψ ψ) 3 ∼ g6 σ 3 f , where σ f is the so-called Hubbard-Stratonovich field, is mapped into the coupling of g 6 tr(ϕ φ) 3 (W (a) 3 of (5.4)); see [69] and [70]. 32In this regard, from the solutions (5.where g6 ↔ g 6 and so, realizing the BF duality with ψ ↔ ϕ 2 , or with ψ ↔ ϕ when including α in the fermion model.In this way, we now examine two new operators: = tr(ψ ψ) tr(F + ij F + ij ); (5.29) with the associated deformations where q = c, d, ..., h from now on.Next, discarding the scalar kinetic term of (5.1), with W (c) 6 , the fermion ψ and gauge A + i equations read respectively, reminding that the second term on the LHS of (5.32) exists when we include both CS terms (for A i and Âi ) in (5.1) and that F − ij = 0, A − i = 0 is set.With just the CS term of (5.2), the ansatz for the U(1) gauge field, with µ, ν for the boundary indices as well and A(r) as another boundary scalar function, gives us a desired solution-see also [5]-as (5.34) In this case, a solution for ψ is read out of (5.10) with a = 0 and ã = i with f (r) in (3.16); and one can also adjust C6 = a 6 ã4 of (3.17), wrt (3.4).However, from the combination of (5.31) and (5.32), we get where taking the third component of the gamma matrices is for compatibility with the solution we take for ψ, which in turn reads from (5.10) with ã = a 1/2 −3ik 16π and so, from (5.31), we have where a 8 = (3πi/k) 1/2 , reminding that F + (r → ∞) → 0. 33 As a result, which, for ∆ + = 6, can be made to correspond to (3.19) with (3.20) and also to (2.13) with an instanton at the conformal point of u = a.Moreover, to confirm the instanton nature of the Euclidean solutions, we compute the value 33 It is noticable that the A + i equation of (5.32) and the solution of (5.10) result in zero magnetic charge or flux, Φ = S 3 ∞ F + = 0; see also [16].
of the corresponding action as where we have used the result of the integral in (5.12) and the same interpretation.
Similarly, for the deformation W (d) 6 of (5.30), discarding the scalar kinetic term of (5.1) and taking both CS terms, the fermion ψ and gauge A + i equations read (5.41) Then, using (5.25), solutions for the fermion and gauge fields are read from (5.10) with ã = a 1/2 −9ik 8π and from (5.37) with a 8 = 1, respectively.However, if we set α = 1 in the equations, a solution for ψ is read from (5.10) with ς = 0 instead of 3/2 along with the gauge solution (5.37) with a 8 = 1/a to have ã the same as before.As a result, we have which corresponds to the bulk near the boundary solution of (3.14) with Č−3 = 0 and Ĉ6 = a 2 ã2 , and of course in the limit of a → 0, r → ∞, reminding the footnote 27.
Another operator we consider is and deform the action of (5.1), discarding its fermion term, with (5.30) with q = e.As a result, the scalar φ equation reads and the gauge A + i equation is the same as (5.17) except leaving out the middle term on the LHS of it.Next, with ϕ = φ, we can take for the gauge part a similar solution to (5.37) with a 8 = 1 and then, taking α ∼ tr(ϕ φ) −3 and F + ∼ h 4 , we obtain a similar solution to (5.7) for h with g 6 = 1 and so, the same correspondence as (5.26) with a = ã for O (e) 6 is confirmed.However, when ϕ = φ we take (5.19) and next, from (5.44) and (5.17) without the middle term on the LHS of the latter, we can write for which we can write a solution as to deform (5.1) with, wrt (5.30), the fermion solution may be (5.10) with a = 0, the scalar solution may be (5.18) and the gauge solution may be F + (r) in (5.34) and A + (r) ∼ 1/r 2 (for O (f ) 6 ) according to the ansatz of (5.8).As a result and a primary test of the correspondence, O (f,g,h) 6 α ∼ 1/r 12 matches with the bulk solution (3.16) and (3.17) for ∆ + = 6 as before.

Summary and Comments
In this article, we started from 11D SUGRA with the background geometry AdS 4 ×S 7 /Z k fixed and a dynamical 4-form ansatz, and got a consistent truncation in that the resulting scalar equations in the external EAdS 4 space do not include any dependence to the internal space ingredients and the associated (pseudo)scalars are H-singlets.In addition, as the solutions are owned to probe (anti)M-branes wrapped around the three internal directions CP 1 ⋉ S 1 /Z k in the (WR)SW background, they break all SUSYs and parity and the resultant theory is for anti-M2-branes.The scale-invariance is also broken due to mass terms and nonlinearities of the equations.Taking the backreaction, we got the massless (m 2 = 0) and massive (m 2 = 1/9, 2/9) modes corresponding to the exactly marginal and marginally irrelevant operators on the 3D boundary and then, wrote a closed solution for the resultant equation and computed its correction to the bulk background action.As well as, for the NPDE equation of the Higgs-like (m 2 = 18) mode, arisen from spontaneous symmetry breaking, we employed the ADM method and arrived at interesting series solutions appropriate for near the boundary analyzes.
In order to realize the supersymmetry and parity breaking and also the realization of H-singlet bulk (pseudo)scalars, we swapped the three fundamental reps of SO (8) and saw that under the branching of G → H such (pseudo)scalars are realized.Because of the bulk symmetries, the boundary duals could be come off in the singlet sectors of ABJM-like models, from which we built some new marginal and irrelevant operators composed of a scalar, a fermion and a U(1) gauge field.Having said that, we saw that solutions with finite actions and SO(4) symmetry on a three-sphere at infinity could be found.After that, we confirmed the state-operator correspondence, adjusted the bulk and boundary parameters and also specified unknown functions in the bulk from the boundary solutions.In addition, we confirmed a type of BF duality (ϕ ↔ ψ) between RB and CF models in terms of the solutions and correspondence.
In order to further confirm the results and reconcile with previous studies by others and their applications, a few more points are worth mentioning.First, we remind that the instantons here are mainly attributed to the unbounded boundary potential from below and have also dual interpretations in the form of Coleman-de Luccia (CdL) bounces [71] mediating falsevacuum decay and the formation of true-vacuum bubble within it; and according to [72], such AdS 4 bubbles collapse and eventually end in a big crunch singularity. 34Second, that our bulk solutions (not) considering the backreaction correspond to (irrelevant) marginally irrelevant deformations (see also the footnote 28) is consistent with the result in [76] that says field theories on dS 3 with SO(3, 1)-invariant solutions and irrelevant deformations are dual to vacuum decays and cosmic singularities in AdS 4 .Third, we notice the probe (anti)M2-branes wrapped around S 3 /Z k that result in domain-walls interpolating among different vacua [77]; and according to [78], a domain-wall at u = 0 separates two degenerate AdS 4 vacua.Indeed, with conformal-invariance breaking, we deal with the problem on constant-u patches, where the boundary is dS 3 in Lorentzian signature.Fourth, such a truncation is interesting in some cosmological (inflationary and bouncing) models 35 ; In fact, our almost degenerate double-well scalar potential from (2.6) accepts bounce solutions and so, it is possible to address the problem from that point of view and provide interesting analyzes.

. 45 )
Then, using the ansatz (5.33) and so = F + = −2 6A(r) + 2 r Á(r) , with a 5 = 1/2 = 5, wrt(3.4),corresponds to the normalizable part of the bulk solution(3.14), with adjusting the constants of both sides.As the same way, with n = 11, it can be made to correspond to f (r) in(3.16).Among other similar operators, if we use any of the following three operators