Higgs-like (pseudo)Scalars in AdS$_4$, Marginal and Irrelevant Deformations in CFT$_3$ and Instantons on $S^3$

With a 4-form ansatz of 11-dimensional supergravity over non-dynamical $AdS_4 \times S^7/Z_k$ background, with the internal space as a $S^1$ Hopf fibration on $CP^3$, we get a consistent truncation. The (pseudo)scalars, in the resulting scalar equations in Euclidean AdS$_4$ space, may be viewed as arising from (anti)M-branes wrapping around internal directions in the (Wick-rotated) skew-whiffed M2-branes background (as the resulting theory is for anti-M2-branes) and so, realizing the modes after swapping the three fundamental representations $\textbf{8}_s$, $\textbf{8}_c$, $\textbf{8}_v$ of $SO(8)$. Taking the backreaction on the external and internal spaces, we get massless and massive modes, corresponding to exactly marginal and marginally irrelevant deformations on the boundary CFT$_3$, and write a closed solution for the bulk equation and compute its correction to the background action. Next, considering the Higgs-like (breathing) mode $m^2=18$, having all supersymmetries, parity and scale-invariance broken, by solving the associated bulk equation with mathematical methods, especially the Adomian decomposition method, and analyzing the behavior near the boundary of the solutions, we realize the boundary duals in $SU(4) \times U(1)$-singlet sectors of the ABJM model. Then, introducing new dual deformation $\Delta_+ = 3, 6$ operators made of bi-fundamental scalars, fermions and $U(1)$ gauge fields, we obtain $SO(4)$-invariant solutions as small instantons on a three-sphere with radius at infinity, which actually correspond to collapsing bulk bubbles leading to big-crunch singularities. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Science and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.