Holography of massive M2-brane theory: non-linear extension

Abstract

We investigate the gauge/gravity duality between the mathcal{N} = 6 mass-deformed ABJM theory with hbox {U}_k(N)times hbox {U}_{-k}(N) gauge symmetry and the 11-dimensional supergravity on LLM geometries with SO(2,1)times SO(4)/{mathbb {Z}}_ktimes SO(4)/{mathbb {Z}}_k isometry, in terms of a KK holography, which involves quadratic order field redefinitions. We establish the quadratic order KK mappings for various gauge invariant fields in order to obtain the canonical 4-dimensional gravity equations of motion and to reduce the LLM solutions to an asymptotically AdS_4 gravity solutions. The non-linearity of the KK maps indicates that we can observe the true purpose of the non-linear KK holography of the LLM solutions. We read the vacuum expectation value of conformal dimension two operator from the asymptotically AdS_4 gravity solutions. For the LLM solutions which are represented by square-shaped Young diagrams, we compare the vacuum expectation value obtained from the holographic procedure with the result obtained from the field theory, which is given by langle mathcal{O}^{(Delta =2)}rangle =sqrt{k}N^{frac{3}{2}}f_{(Delta =2)}+mathcal{O}(N), where f_{Delta } is independent of N. Based on this result, we examine the gauge/gravity duality in the large N limit and finite k. We also show that the vacuum expectation values of the massive KK graviton modes are vanishing as expected by the supersymmetry.