Marginal fluctuations as instantons on M2/D2-branes

We introduce some (anti-) M/D-branes through turning on the corresponding field strengths of the 11- and 10-dimensional supergravity theories over $$\hbox {AdS}_4 \times M^{7|6}$$ spaces, where we use $$S^7/Z_k$$ and $$CP^3$$ for the internal spaces. Indeed, when we add M2/D2-branes on the same directions with the near horizon branes of the Aharony–Bergman–Jafferis–Maldacena model, all symmetries and supersymmetries are preserved trivially. In this case, we obtain a localized object just in the horizon. This normalizable bulk massless scalar mode is a singlet of $$SO(8)$$ and $$SU(4) \times U(1)$$, and it agrees with a marginal boundary operator of the conformal dimension of $$\Delta _+=3$$. However, after performing a special conformal transformation, we see that the solution is localized in the Euclideanized $$\hbox {AdS}_4$$ space and is attributable to the included anti-M2/D2-branes, which are also necessary to ensure that there is no back-reaction. The resultant theory now breaks all $${\mathcal {N}}=8,6$$ supersymmetries to $${\mathcal {N}}=0$$, while the other symmetries are so preserved. The dual boundary operator is then set up from the skew-whiffing of the representations $$\mathbf 8 _s$$ and $$\mathbf 8 _v$$ for the supercharges and scalars, respectively, while the fermions remain fixed in $$\mathbf 8 _c$$ of the original theory. Besides, we also address another alternate bulk to boundary matching procedure through turning on one of the gauge fields of the full $$U(N)_k \times U(N)_{-k}$$ gauge group along the same lines with a similar situation to the one faced in the AdS$$_5$$/CFT$$_4$$ correspondence. The latter approach covers the difficulty already faced with in the bulk–boundary matching procedure for $$k=1,2$$ as well