Dual localized objects from m-branes over ${{AdS}}_{4}\times {S}^{7}/{Z}_{k}$

Abstract

We consider a few ansatzs for the four and seven forms of 11-dimensional supergravity over AdS 4 ×S 7 /Z k while try to keep the geometry unchanged. From the 4-form equations, we arrive at some massless scalars and pseudoscalars in the bulk of Euclidean AdS 4 that match with some boundary ∆ + = 3 operators. Indeed, the main objects are instantons and domain walls as the fully and partially localized objects in the external space, respectively. The latter comes from an (anti)M5-brane wrapping partly around three internal directions similar to the fuzzy S 3 /Z k solutions. Except for the first solutions of adding some anti-M2-branes|(M2-branes) to the original M2-branes, the objects backreact on the geometry although small|(break all supersymmetries and destabilize the vacua). The dual 3-dimensional field theory solutions are got by the skew-whiffing 8 s → 8 v and 8 s → 8 c for the scalars and pseudoscalars respectively, while the gauge fields are used mainly for the k = 1, 2 cases where the R-symmetry and supersymmetry are enhanced as SU(4) × U(1) → SO(8) and N = 6 → N = 8 respectively, and also for pseudoscalars. Further, for the pseudoscalars we propose a special boundary deformation, with a fermion field, that is equivalent to a multi-trace deformation already studied for the bulk m 2 = −2 conformally coupled pseudoscalar. *

Massive (pesudo) scalars in AdS4, SO(4) invariant solutions and holography

“…where b is a boundary constant and A, as a normalization factor, comes from solving the coupled equations as A = a/ √ 3 and note that α = tr(ψψ) −1 . Still, from satisfying the U (1) gauge equation, one could see that the magnetic charge because of A + vanishes [2]. In addition, from this solution, one can confirm the state-operator correspondence in leading order according to (5.3).…”

“…On the other hand, although the original ABJM model has even parity, to make a dual for the non-minimal bulk pseudoscalar with parity breaking scheme, we keep just one part of the quiver gauge group next to its CS term together with massless matter fields. This aim is realized through fractional M2-branes idea [10] and the Novel Higgs-mechanism [27], which we already employed in [2], with a focus on just an U (1) part of U (1) × U (1) in that our pseudoscalars are neutral under A + i ≡ (A i +Â i ), and set A − i ≡ (A i −Â i ) = 0. Then, with the ansatz ψ â a = δ â a N ψ for the singlet fermion and just one scalar with Y = h(r) N I N , where h(r) is a scalar function on the boundary and I N is the unit N × N matrix 6 , from the main ABJM action [8], the scalar and fermion potentials vanish and so, we can write…”

“…In this subsection, as an example, we make a solution based on this single-trace deformation for one of the operators. For ∆ + = 4, it might be considered as a double-trace deformation (see, for instance, [30] and references therein) of O − 2 = ψψ of the conformally coupled pseudoscalar m 2 R 2 AdS = −2 already studied in [2]. For ∆ + = 5, it may be considered as…”

“…where the latter operator is indeed tr(X [I X J] ) whose details were studied in [31] and we have recently considered a special version of it in [2] as O + 1 = 3 4 tr(yȳ). Finally, for ∆ + = 6, we can consider the operator as…”

“…In the past few years, we have studied nonperturbative and localized objects in M2-and D2brane theories; see [1] and [2] as our recent studies. The focus was on Instantons, as classical solutions to Euclidean Equations of Motion (EOM) with finite actions, which contribute to phase integrals and mediate tunneling among vacua.…”