Instantons in AdS$_4$ from (anti)membranes wrapping $S^7$ to Bose–Fermi duality in CFT$_3$’s

Abstract

We present new $\mathrm{SO}(4)$-invariant and non-supersymmetric instanton solutions for the conformally coupled $m^2=-2$ and massive $m^2=+4$ (pseudo)scalars arising from a consistent truncation of 11-dimensional supergravity over $\mathrm{AdS}_4 \times S^7/Z_k$ when the internal space is a $S^1$ Hopf fibration on $\mathrm{CP}^3$, and we consider backreaction. In fact, the bulk configurations associate with (anti)membranes wrapped around mixed internal (and external) directions, which in turn probe the Wick-rotated or skew-whiffed background, break all supersymmetries as well as parity invariance. From near the boundary behavior of the closed solution for the coupled bulk (pseudo)scalar, we get a marginal triple-trace deformation with mixed boundary condition (valid also for the bulk massless $m^2=0$ (pseudo)scalar, raised when considering the external space backreaction, with Dirichlet boundary condition) and as a result, the corresponding boundary effective potential is unbounded from below and causes an instability because of the Fubini-like instanton. Presenting dual effective actions, we see that the boundary solutions and counterparts realize in singlet sectors of three-dimensional U(N) and O(N) Chern–Simons-matter field theories. In particular, we use versions of massless and mass-deformed regular and critical boson and fermion models, find instantons and confirm state-operator AdS$_4$/CFT$_3$ correspondence and also Bose–Fermi duality at the level of the solutions. In addition, we discuss on relations of our setups with Vasiliev’s Higher-Spin theories, deformations of the Aharony–Bergman–Jafferis–Maldacena model and other related studies.

Appendices

Appendix A: 11D Supergravity action and equations

In this study, we use the Euclidean version of the bosonic part of the 11D SUGRA action as

$$\begin{aligned} S_{11}^E = -\frac{1}{2 \kappa _{11}^2} \left[ \int d^{11}x \, \sqrt{g} \, {\mathcal {R}} + \frac{1}{2} \int \left( {G}_4 \wedge *_{11} {G}_4 - \frac{i}{3}\, {{\mathcal {A}}}_3 \wedge {G}_4 \wedge {G}_4 \right) \right] , \end{aligned}$$

(A.1)

where, in general, $2 \kappa _{D}^2 = 18 \pi {\mathcal {G}}_{D}=\frac{1}{2\pi } (2\pi l_p)^9$, with $\kappa _{D}$, ${\mathcal {G}}_{D}$ and $l_p$ as the D-dimensional gravitational constant, Newton’s constant and Plank length, respectively; and $G_4= d{\mathcal {A}}_3$.

The resultant EoMs for ${\mathcal {A}}_3$ and $g_{MN}$ read

$$\begin{aligned}&d *_{11} G_4 -\frac{i}{2}\, G_4 \wedge G_4 \equiv d {\hat{G}}_7 =0, \end{aligned}$$

(A.2)

$$\begin{aligned}&{\mathcal {R}}_{MN} - \frac{1}{2} g_{MN}\, {\mathcal {R}} = 8 \pi \, {\mathcal {G}}_{11}\, T_{MN}^{{G}_4}, \end{aligned}$$

(A.3)

with $M, N,\ldots$ for the 11D space-time indices and

$$\begin{aligned} T_{MN}^{{G}_4} = \frac{1}{4!} \left[ 4\, {G}_{MPQR}\, {G}_N^{PQR} - \frac{1}{2} g_{MN}\, {G}_{PQRS}\, {G}^{PQRS} \right] , \end{aligned}$$

(A.4)

respectively.

It is notable that the bosonic fields of 11D SUGRA include the graviton $g_{MN}$ with 44 degrees of freedom and the rank-3 antisymmetric tensor ${\mathcal {A}}_3$ with 84 degrees of freedom, so that the total 128 degrees of freedom equal to degrees of freedom of the gravitino $\Psi _M$, which in turn is the only necessary fermion field (a 32-component Majorana spinor) in the theory to preserve supersymmetry. Beside the extended objects, coupled to the gauge field ${\mathcal {A}}_3$ and its 11D dual, which are (electrically charged) M2-branes and (magnetically charged) M5-branes, there are also M9-branes and a pair of purely gravitational objects namely gravitational wave ($M {\mathcal {W}}$) and Kaluza–Klein (${\mathcal {K}} {\mathcal {K}}$) monopoles. In addition, the electrical or page charge and magnetic or topological charge

$$\begin{aligned} Q_e = \frac{1}{\sqrt{2}\, \kappa _{11}^2} \int {\hat{G}}_7, \quad Q_m=\frac{1}{\sqrt{2}\, \kappa _{11}^2} \int G_4, \end{aligned}$$

(A.5)

obeying the Dirac quantization condition $Q_e Q_m = 2 \pi Z$, come from the ${\mathcal {A}}_3$ equation (A.2) and the Bianchi identity $dG_4=0$, respectively.

Further, having a classical solution, the killing spinors $\epsilon$ control the numbers of supersymmetries from zeroing the gravitino variation

$$\begin{aligned} \delta \Psi _M = D_M \epsilon -\frac{1}{128} \left( \Gamma _M^{PQRS} - 8\, \delta _M^P\, \Gamma ^{QRS} \right) \, G_{PQRS}\, \epsilon =0, \end{aligned}$$

(A.6)

where $D_M$ is for the covariant derivative and $\Gamma ^{M_1 M_2 \ldots }$ are the anti-symmetrized higher dimensional gamma matrices.

Appendix B: Matching with setups and results of [20]

As we mentioned at the end of Sect. 2.1, a general truncation of 11D SUGRA using the anstazs

$$\begin{aligned} ds_{11}^2&= ds_4^2 + e^{2 U} ds^2 ({\mathrm{KE}_6}) + e^{2 V} \left( \eta + A_1 \right) ^2, \end{aligned}$$

(B.1)

$$\begin{aligned} {\tilde{G}}_4&= f\, \texttt {vol}_4 + H_3 \wedge \left( \eta + A_1 \right) + H_2 \wedge J + H_1 \wedge J \wedge \left( \eta + A_1 \right) + 2 h\, J \wedge J \nonumber \\&\quad + \sqrt{3}\, \big [\chi _1 \wedge \Omega + \chi \, \left( \eta + A_1 \right) \wedge \Omega + c.c. \big ] \end{aligned}$$

(B.2)

is worked out in [20], where $ds_4^2$ and $ds^2 ({\mathrm{KE}_6})$ (a Kähler–Einstein metric) are the full metrics of $\mathrm{AdS}_4$ and $\mathrm{CP}^3$ spaces respectively, U, V, h, f are scalar fields, $A_1$ is an 1-form, $H_p\, (p=1,2,3)$ are p-forms, $\chi _1$ is a complex 1-form and $\chi$ is a complex scalar on the external 4D space; the (3, 0)-form $\Omega$ defines the complex structure on $\mathrm{CP}^3$ and “c.c.” denotes complex conjugate.

Given the above ansatzs, the details of KK reduction, after solving the 11D SUGRA equations, are given in Appendix B of [20]. By comparing the results there and our results, we see that the solutions match with $U = V = \chi = A_1 = B_1 = B_2 = 0$ (noting that $H_1=dh,\, H_2=dB_1+2B_2 +hF_2,\, H_3=dB_2,\, F_2=dA_1,\, \chi _1=-\frac{i}{4} d\chi - A_1 \chi$), $\eta =2 e_7$, $\texttt {vol}_4 =\frac{R^4}{16} {\mathcal {E}}_4$ and

$$\begin{aligned}&f = 6\, f_1, \quad h= 4\, R^4 f_3, \quad dh= - R^4\, df_2, \end{aligned}$$

(B.3)

$$\begin{aligned}&f = \frac{6}{R^7} \left( \epsilon + h^2 \right) , \quad \epsilon =\pm \, C_3\, R^6, \end{aligned}$$

(B.4)

where the last equation comes from (B.12) of [20]. In particular, from the equation (B.11) there, we read

$$\begin{aligned} *_4 d (*_4 dh)- (16 + 24 \epsilon )\, h - 2 \times 12\, h^3 =0, \end{aligned}$$

(B.5)

which is indeed the equation for $f_3$ in (2.5) up to some scaling and considering that $R=1$ is set; and also note that with $G_4 \rightarrow i G_4$, the Euclidean equation (A.2) goes to that in [20]. It is also noticeable that from the latter equation, we get the modes $m^2 R_{\mathrm{AdS}}^2=-2$ with $\epsilon = -1$ (SW) and $m^2 R_{\mathrm{AdS}}^2=10$ with $\epsilon = 1$ (WR), which were recently discussed in [4, 6] respectively.

It is also noticeable that the 4D Eq. (B.5) for h can also be obtained from the string-frame action (2.10) of [20], which with our adjustments reads

$$\begin{aligned} S_{4E} = \int d^4x\, \sqrt{g_4}\, R^7 \left[ - \left( {\mathcal {R}}_4-{\mathcal {X}} \Lambda \right) + \frac{3}{2 R^6} \left( \nabla h \right) ^2 + \frac{3 h^2}{R^8} \left( 8+ \frac{12 \epsilon }{R^6}+ \frac{6 h^2}{R^6}\right) \right] , \end{aligned}$$

(B.6)

where ${\mathcal {R}}_4$ is the scalar curvature of $\mathrm{EAdS}_4$ and ${\mathcal {X}} \Lambda = \frac{1}{R^2} \left( -42+ 18 \epsilon ^2 \right)$ with $\Lambda$ as the cosmological constant, noting that for $\epsilon = \pm 1$ there is the well-known result ${\mathcal {X}}=2$.

On the other hand, with the 11D Einstein equations

$$\begin{aligned} {\mathcal {R}}_{MN}= \frac{1}{12} {G}_{MPQR}\, {G}_N^{PQR} - \frac{1}{144} g_{MN}\, {G}_{PQRS}\, {G}^{PQRS}, \end{aligned}$$

(B.7)

for the components ${\mathcal {R}}_{\mu \nu }$, ${\mathcal {R}}_{mn}$ and ${\mathcal {R}}_{77}$, the equations (B.19), (B.21) and (B.22) of [20] follow respectively, which in turn correspond to the Eqs. (2.8), (2.10) and (2.9), when written for $f_3$ and the above adjustments are considered. Finally, as argued also in [20], becuase all dependencies on the internal 7D space is dropped out of the 11D equations and we are left with the equations for the 4D fields, therefore the ansatzs (2.1) and (2.2) defines a consistent Kaluza–Klein truncation.

Appendix C: To compute the instanton (3.4) correction to the background action

Here we compute the correction to the original action because of the bulk instanton solution (3.4). As the background geometry is unchanged, the right parts of the bosonic 11D SUGRA action (A.1) for our purpose, are the second and third terms.

On the other hand, from the ansatz (2.1), we get the 11D dual 7-form

$$\begin{aligned} {G}_7 = R^7\, {f}_1\, J^3 \wedge e_7 + R^5\, *_4 df_2 \wedge J^2 + R^7\, f_3\, {\mathcal {E}}_4 \wedge J \wedge e_7; \end{aligned}$$

(C.1)

and write

$$\begin{aligned} {G}_4 = d{{\mathcal {A}}}_3, \quad {{\mathcal {A}}}_3 = \tilde{{\mathcal {A}}}_3^{(0)} + \left( 8\, R^8 \right) \left( f_3\, J \wedge e_7 \right) , \quad {\tilde{G}}_4^{(0)} = d\tilde{{\mathcal {A}}}_3^{(0)} = \frac{3}{8} R^4 f_1\, {\mathcal {E}}_4. \end{aligned}$$

(C.2)

With these at hand, using (2.3) and (2.4) and after some math manipulations, we arrive at

$$\begin{aligned} {\tilde{S}}_{11}^E = - \frac{R^{11}}{32\, \kappa _{11}^2} \int \bigg [-3\, c_3^2 + R^2\, (\partial _{\mu } f_2)(\partial ^{\mu } f_2)+ 4\, f_2^2 + 12\, R^2\, f_2^4 \bigg ]\, {\mathcal {E}}_4 \wedge J^3 \wedge e_7, \end{aligned}$$

(C.3)

where we have discarded the surface term of $d\left( f_2^2\, {\mathcal {A}}_3^{0} \right)$ in the integrand that, as a total derivative, does not contribute to the equations; and that $\mu , \nu ,\ldots$ are $\mathrm{AdS}_4$ space indices. It is also important to note that the mode $m^2 R_{\mathrm{AdS}}^2=-2$, which we are interested in its associated solution (3.4) here, is also realizable in the SW background $C_2=0, C_3=1$ of (2.7)²⁰ and so, we use the same setting for the Eq. (2.12) stemmed from combining the Eqs. (2.6) and (2.10).

To continue computing (C.3), we use

$$\begin{aligned}&\texttt {vol}_4 = \frac{R^4}{16} \int {\mathcal {E}}_4, \quad {\mathcal {E}}_4 = -\frac{du}{u^4} \wedge dx \wedge dy \wedge dz, \end{aligned}$$

(C.4)

$$\begin{aligned}&\texttt {vol}_7 = \frac{R^7}{3!} \int J^3 \wedge e_7= \frac{\pi ^4\, R^7}{3\, k}, \end{aligned}$$

(C.5)

where $\texttt {vol}_4$ and $\texttt {vol}_7$ are the full external and internal volumes, and

$$\begin{aligned} {R}/{l_p}= \left( k\, N\, 2^5 \pi ^2 \right) ^{1/6}, \quad 4 \pi \kappa _{11}^2 = \left( 2 \pi \, l_p \right) ^9 \Rightarrow \kappa _{11}^2 = \frac{16}{3} \pi ^5 \left( \frac{R^9}{3\, k^3} \right) ^{1/2}, \end{aligned}$$

(C.6)

where the first relation from the left is from [5].

Then, we note that the first term on the RHS of (C.3) is for the SW background realized with ${C}_3=1$, and to compute the remaining part, we use the instanton solution (3.4) (with $C_2=0$) with the 3D spherical coordinates.

Appendix D: Solutions for the main equation with $m^2=-2, +4$

For the CC (pseudo)scalar $m^2=-2$, with the parameters $C_3=\frac{13}{12}$, $C_2=\frac{1}{24\, \sqrt{2}}$, in the SW version of the main Eq. (2.6), the resultant equation reads

$$\begin{aligned} \Box _4\, f + 2\, f + 3\sqrt{2}\, f^2 - 24\, f^3 = 13/(72 \sqrt{2}). \end{aligned}$$

(D.1)

To have an approximate solution to the equation, besides the one based on the closed solution in [6], we rewrite the homogeneous part of the Eq. (D.1) (discarding the non-homogenous F term, which of course adds just a non-dynamical or constant term to the final solution), making use of the conformal flatness of the Euclidean AdS space and the scaling of (3.3), as

$$\begin{aligned} \left( \partial _i \partial _i + \partial _u \partial _u \right) \, g_{i+1} = \sum _{i=0}^{\infty } A_i, \end{aligned}$$

(D.2)

with

$$\begin{aligned} A_0 = -\frac{\delta }{u} g_0^2 + 24\, g_0^3 , \quad A_1= -\frac{2 \delta }{u} g_0\, g_1 + 72\, g_0^2\, g_1, \ldots , \end{aligned}$$

(D.3)

and the leading-order (LO) solution

$$\begin{aligned} \left[ \frac{\partial ^2}{\partial r^2} + \frac{2}{r} \frac{\partial }{\partial r} + \frac{\partial ^2}{\partial u^2} \right] g_0(u,r)=0 \Rightarrow g_0(u,r) =\frac{{b}_0\, u}{\left[ (u+a_0)^2 + r^2 \right] }, \end{aligned}$$

(D.4)

where $r= \sqrt{x^2+y^2+z^2}$ with discarding the angular parts of the 3D spherical Laplacian for simplicity, and note that ${b}_0$ here is an arbitrary constant. Then, if we use the series expansion of the solution (D.4) about $u=0$ as

$$\begin{aligned} {\bar{g}}_0(u,r) =\frac{{b}_0}{(a_0^2+ r^2)} \left[ 1 - \frac{2\, a_0}{(a_0^2 + r^2)}\, u \right] , \quad {\bar{f}}_0(u,r)= {\bar{g}}_0(u,r)\, u, \end{aligned}$$

(D.5)

as the initial data in the recursion of the Eq. (D.2), we get the approximate solution

$$\begin{aligned} f^{(2)}(u,r)= {\bar{f}}_0(u,r) - \frac{3\, {b}_0^2}{(a_0^2 + r^2)^2}\, \left[ \frac{{b}_0}{a_0} -\sqrt{2}\, (1-\ln (u)) \right] \, u^2+ O(u^3), \end{aligned}$$

(D.6)

up to the second-order of the perturbation.

Similarly, another way to realize the massive (pseudo)scalar $m^2=4$ is with $C_3=1, C_2=\frac{\sqrt{3}}{12}$, in the SW version of (2.6) as

$$\begin{aligned} \Box _4\, f - 4\, f + 12\sqrt{3}\, f^2 - 24\, f^3 = 0, \end{aligned}$$

(D.7)

where $F=0$ is realized interestingly.

Then, to write a solution for the Eq. (D.7), one may use the so-called Witten’s solution [95], for its linear part in coordinate-space, as

$$\begin{aligned} f_0(u,\vec {u})= \frac{8}{\pi ^{2}} \left[ \frac{u}{u^2+(\vec {u}-\vec {u}_0)^2} \right] ^{4} \Rightarrow f_0(u\rightarrow 0,r)= \frac{8}{\pi ^{2}} \left( \frac{u}{r^2} \right) ^4 + O(u^6), \end{aligned}$$

(D.8)

where we have rewritten its behavior near the boundary on the right as well; and then, one can use perturbative methods to arrive at higher-order expansions around the LO solution.

On the other hand, we can use the self-similar reduction method (see, for instance, [96]), to solve the main NPDE (2.6) through the scale-invariance of the variables

$$\begin{aligned} u\rightarrow S\, u, \quad r \rightarrow S^{s_1}\, r, \quad f\rightarrow S^{s_2}\, f, \quad \Rightarrow f(u,r)\rightarrow u^{s_3}\, {\mathcal {F}}(\xi ), \quad \xi = {r}\,{u^{s_4}}, \end{aligned}$$

(D.9)

where S is the scaling-parameter, $s_1=-s_4=1$ and $s_2=s_3=0$ in this case. Doing so, the resultant NODE (for the SW version with $C_3\ge 1/3$) reads

$$\begin{aligned} \left[ \left( {\xi }^{2}+1 \right) {\frac{{d }^{2}}{{d}{\xi }^{2}}} + \frac{(2+ 4\, \xi ^2)}{\xi }\,\frac{d}{d \xi }- m^2 \right] {\mathcal {F}}(\xi ) + 6 \sqrt{3}\, m\, {\mathcal {F}}(\xi )^{2} - 24\, {\mathcal {F}}(\xi )^{3}=0; \end{aligned}$$

(D.10)

Then, a solution for its linear part becomes

$$\begin{aligned} {\mathcal {F}}_0(\xi ) = \frac{1}{\xi } \left[ C_6\, P_{\Delta _+ - 2}(i \xi ) + C_7\, Q_{\Delta _+ - 2} (i \xi ) \right] , \end{aligned}$$

(D.11)

where $P_a(Z)$ and $Q_a(Z)$ are for the Legendre functions of the first- and second-kind, respectively. Now, taking the LO solution $F_0$ in palace of the function in nonlinear terms of (D.10), we get a first-order solution in Legendre functions, again. Alternatively, one may also note that the Eq. (D.10) without the nonlinear terms is indeed the Riccati equation, and that for special bulk modes, its solutions are in terms of inverse trigonometric functions.²¹ Then, if we use the latter solutions, for a special bulk mode, to obtain ${\mathcal {F}}_1(\xi )$, the suitable terms (corresponding to the operators replying the bulk modes by the standard AdS/CFT dictionary) of the perturbative solution up to the first-order (noting ${\mathcal {F}}={\mathcal {F}}_0 + {\mathcal {F}}_1 +\cdots$), after substitution $\xi ={r}/{u}$ and series expansion around $u=0$, is (3.8).

Appendix E: Basics of AdS$_4$/CFT$_3$ dictionary for (pseudo)scalars

For a (pseudo)scalar in Euclidean $\mathrm{AdS}$, near the boundary ($u=0$) behavior reads

$$\begin{aligned} f(u\rightarrow 0, \vec {u}) \approx u^{\Delta _-} \left( \alpha (\vec {u})+\cdots \right) + u^{\Delta _+} \left( \beta (\vec {u}) + \cdots \right) , \end{aligned}$$

(E.1)

where $2\, \Delta _{\mp }=3 \mp \nu$ with $\sqrt{9 + 4\, m^2}=2 \nu$ in $\mathrm{AdS}_4$. Such a scalar could be quantized with the standard or Dirichlet ($\delta \alpha =0$), alternate or Neumann ($\delta \beta =0$) [95, 97–99] or Mixed boundary conditions [29]—see also [100]—and that the latter BC could be considered as a multi-trace deformation of the field theory dual to the Neumann BC [101]. Indeed, for the (pseudo)scalars, the standard BC is used for any mass while regularity ($\Delta _+$ being real) and stability need satisfying the Breitenlohner–Freedman (BF) bound $m^2\ge m_{\mathrm{BF}}^2=-9/4$ [102]; and at the same time, the alternate BC is always used for $-9/4 \le m^2 \le -5/4$ ensuring stability as well [103]. We also remind that $\alpha$ and $\beta$ have holographic descriptions as source and vacuum expectation value (vev) of the one-point function of the operator $\Delta _+$ respectively, and conversely for the operator $\Delta _-$. Besides, with $m^2\ge -5/4$, only the $\beta$ mode is normalizable and so, dual operator with $\Delta _+$ and Dirichlet BC should be used; while for the masses in the range lower than the latter value and upper than the BF bound, both (now normalizable) modes and BCs are allowed.

For the standard and alternate boundary conditions, we use the following Euclidean AdS/CFT dictionary:

$$\begin{aligned} \begin{aligned}&\sigma \equiv \langle \mathcal {{O}}_{\Delta _+} \rangle _{\alpha } = - \frac{\delta W[\alpha ]}{\delta \alpha } = \frac{1}{3} \beta , \quad \langle \mathcal {{O}}_{\Delta _-} \rangle _{\beta } = - \frac{\delta {\tilde{W}}[\sigma ]}{\delta \sigma }= \alpha , \\&{\tilde{W}}[\sigma ] = - W[\alpha ] - \int d^3 \vec {u}\ \alpha (\vec {u})\, \sigma (\vec {u}), \\&W[\alpha ] = -S_{\mathrm{on}}[\alpha ]=\Gamma _{\mathrm{eff.}} [\alpha ], \quad {\tilde{W}}[\sigma ] = - {\tilde{S}}_{\mathrm{on}}[\sigma ]= {\tilde{\Gamma }}_{\mathrm{eff.}} [\sigma ], \end{aligned} \end{aligned}$$

(E.2)

where $S_{\mathrm{on}}$ and ${\tilde{S}}_{\mathrm{on}}$ are for the bulk $\mathrm{AdS}_4$ on-shell actions, and $W[\alpha ]$ and ${\tilde{W}}[\sigma ]$ (Legendre transform of the other) are respectively generating functionals of the connected correlators of $\mathcal {{O}}_{\Delta _+}$ and $\mathcal {{O}}_{\Delta _-}$ of the boundary CFT$_3$, and also $\Gamma _{\mathrm{eff.}} [\alpha ]$ and ${\tilde{\Gamma }}_{\mathrm{eff.}} [\sigma ]$ (Legendre transform to each other) are respectively the effective actions of dual CFT (with $\Delta _-$ quantization) and usual CFT (with ${\Delta _+}$ quantization).

Still, one may consider deformations of the boundary CFT by a function ${V}(O_{\mp })$ of the local operators; see, for instance, [29, 104–107]. In particular, for the mixed BC, also considered as a multi-trace deformation of the boundary theory dual to the Neumann BC [101], we can write

$$\begin{aligned} \begin{aligned}&S_{\mathrm{on}}^f[\alpha ] = S_{\mathrm{on}}[\alpha ] + \int d^3 \vec {u}\ {V}(\alpha ) \Rightarrow \sigma _f=\sigma - \acute{{V}}(\alpha ), \\&\Gamma _{\mathrm{eff.}}^f [\alpha ]= \Gamma _{\mathrm{eff.}}[\alpha ] + \int d^3 \vec {u}\ {V}(\alpha ), \end{aligned} \end{aligned}$$

(E.3)

where the f index is for the associated quantity after the deformation, the prime on V is for derivative wrt its argument $(\alpha )$, and in particular $\sigma _f$ is the new source for the operator $\mathcal {{O}}_-$ with the vev of $\alpha$. It is also mentionable that exchanging $\sigma \leftrightarrow \alpha$ in the above relations (E.3) is valid for deformations with the Dirichlet BC.

Appendix F: Holographic actions for the bulk CC (pseudo)scalar and interpretations

For a free minimally coupled bulk (pseudo)scalar in the fixed or non-dynamical $\mathrm{AdS}$ background, like that in (2.12), one can compute the boundary effective actions corresponding to the three boundary conditions and rules of the Sect. 4 and particularly the Appendix E; see [101, 107]. In particular, the regularized and renormalized effective actions corresponding to the Dirichlet and Neumann or mixed boundary conditions, using the Hamiltonian–Jacobi method and in two-derivative approximation, are computed in [101], for the CC case with $\Delta _{\pm }=2,1$, to be

$$\begin{aligned}&\tilde{\Gamma }_{\mathrm{eff.}} [\sigma ] \approx \int d^3\vec {u}\ \left[ \frac{1}{2} \sigma ^{-3/2}\, (\partial _i \sigma ) (\partial ^i \sigma ) + \sigma ^{1/2}\, \mathcal {R}_3 \right] , \end{aligned}$$

(F.1)

$$\begin{aligned}&{\Gamma }_{\mathrm{eff.}}^f [\alpha ] = \int d^3\vec {u}\ \left[ \frac{\alpha ^{-1}}{8} (\partial _i \alpha ) (\partial ^i \alpha ) + \frac{\alpha }{16} \mathcal {R}_3 + \tilde{c}_0\, \alpha ^3 + V(\alpha ) \right] , \end{aligned}$$

(F.2)

respectively, where ${\mathcal {R}}_3=6/R_0^2$ is the scalar curvature of the three-sphere $S^3$ (with $R_0\rightarrow \infty$ for the flat $R^3$ space or $S^3$ at infinity: $S^3_\infty$).

Then, we first note to the solution (3.4) for the case at hand that, with (3.6) and the rules (E.2) and (E.3), indeed corresponds to a triple-trace deformation of the boundary theory with

$$\begin{aligned} V(\alpha ) = -\frac{1}{3} {\hat{h}}\, \alpha ^3, \quad {\hat{h}}= \frac{2 a_0}{b_0}; \end{aligned}$$

(F.3)

²²Therefore, from the last two terms of (F.2) and from (F.3), we can write the holographic effective potential

$$\begin{aligned} V_{\mathrm{eff.}}(\alpha ) =\frac{1}{3} \left( {\hat{h}}_0 - {\hat{h}} \right) \alpha ^3, \end{aligned}$$

(F.5)

where ${\hat{h}}_0=3 {\tilde{c}}_0$; and it is argued in [101] that for the 4D ${\mathcal {N}}=8$ gauged supergravity, ${\tilde{c}}_0=0$, where the bulk solution describes a sector of the Coulomb branch of the dual boundary theory.^23,24

In particular, it is shown in [107] that for ${\hat{h}}> {\hat{h}}_0$, there is a runway solution where the vev of the operator ${\mathcal {O}}_1 (=\alpha )$ diverges in finite time (indeed, the rolling down of the scalar from the negative potential and producing a big-crunch at finite time) and the associated CFT$_3$ needs an UV-completion. In other words, to define a field theory, one must introduce an UV cut-off $\Lambda$ as the boundary relevant operator behaves badly at UV; and that (for $g_6 > g_6^c$; see below) the UV cut-off $\Lambda$ is not removed and an UV-completion is required to have a cut-off independent theory; and in this case, there is a big-crunch singularity in the bulk. In simpler language, the deformation or negative potential (F.5) is unbounded from below and causes instability.²⁵

In fact, this setup was originally considered about 3D O(N) vector models; see also [9, 113]. In this way, we may take $g_6^c=2\, {\hat{h}}_0$ and $g_6=2\, {\hat{h}}$, and note that in the classical approach, the sign of $g_6$ determines the system behavior and for the quantum case, being larger or smaller than $g_6^c=(4 \pi )^2$ [65]; and that for $g_6 < g_6^c$, the tri-critical model is scale-invariant in LO, but above the critical value $g_6^c$, a nonzero mass parameter appears (i.e., $m_b^2\, \langle y {\bar{y}}\rangle$, with y as the singlet boundary scalar)²⁶ and so, the SI is broken because of 1/N corrections—for further related discussions, see [115, 116], where the SI breaking in 3D U(N) plus CS theory is discussed, and also [117]. On the other hand, in the Large-N limit, non-perturbative effects destabilize the model for $g_6 > g_6^c$ and the boundary configurations decay in a finite time because of an infinite family of instantons on $S^3$ of the radius $R_0$- In other words, there is tunneling from the local minimum at $\alpha =0$ to the instability region at $\alpha \rightarrow \infty$. It should also be mentioned that for $g_6 > g_6^c$, only the neutral bounded states under U(1), which in turn belong to neutral irreducible reps of $\mathrm{SU}(N)$, are remained [118].