Marginal fluctuations as instantons on M2/D2-branes

Abstract

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.