We test the holographic relation between the vacuum expectation values of gauge invariant operators in N = 6 U k (N ) × U −k (N ) mass-deformed ABJM theory and the LLM geometries with Z k orbifold in 11-dimensional supergravity. To do so, we apply the Kaluza-Klein reduction to construct a 4-dimensional gravity theory and implement the holographic renormalization procedure. We obtain an exact holographic relation for the vacuum expectation values of the chiral primary operator with conformal dimension = 1, which is given by, for large N and k = 1. Here the factor f ( ) is independent of N . Our results involve an infinite number of exact dual relations for all possible supersymmetric Higgs vacua and so provide a non-trivial test of gauge/gravity duality away from the conformal fixed point. We extend our results to the case of k = 1 for LLM geometries represented by rectangularshaped Young diagrams. We also discuss the exact mapping of the gauge/gravity at finite N for classical supersymmetric vacuum solutions in field theory side and corresponding classical solutions in gravity side.

The duality between a d-dimensional conformal field theory with relevant deformation and a gravity theory on an asymptotically AdS d+1 geometry, has become a suitable tool in the investigation of the emergence of gravity from quantum entanglement in field theory. Recently, we have tested the duality between the mass-deformed ABJM theory and asymptotically AdS 4 gravity theory, which is obtained from the KK reduction of the 11-dimensional supergravity on the LLM geometry. In this paper, we extend the KK reduction procedure beyond the linear order and establish non-trivial KK maps between 4-dimensional fields and 11-dimensional fluctuations. We rely on this gauge/gravity duality to calculate the entanglement entropy by using the Ryu-Takayanagi holographic formula and the path integral method developed by Faulkner.We show that the entanglement entropies obtained using these two methods agree when the asymptotically AdS 4 metric satisfies the linearized Einstein equation with nonvanishing energy-momentum tensor for two scalar fields. These scalar fields encode the information of the relevant deformation of the ABJM theory. This confirms that the asymptotic limit of LLM geometry is the emergent gravity of the quantum entanglement in the mass-deformed ABJM theory with a small mass parameter. We also comment on the issue of the relative entropy and the Fisher information in our setup.

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We present a detailed account and extension of our claim in arXiv:1610.01490. We test the gauge/gravity duality between the N = 6 mass-deformed ABJM theory with U k (N )×U −k (N ) gauge symmetry and the 11-dimensional supergravity on LLM geometries with SO(4)/Z k ×SO(4)/Z k isometry, in the large N limit. Our analysis is based on the evaluation of vacuum expectation values of chiral primary operators from the supersymmetric vacua of mass-deformed ABJM theory and from the implementation of Kaluza-Klein holography to the LLM geometries. We focus on the chiral primary operator with conformal dimension ∆ = 1. We show that O (∆=1) = N 3 2 f (∆=1) for all supersymmetric vacuum solutions and LLM geometries with k = 1, where the factor f (∆) is independent of N . We also confirm that the vacuum expectation value of the energy momentum tensor is vanishing as expected by the supersymmetry. We extend our results to the case of k = 1 for LLM geometries represented by rectangular-shaped Young-diagrams. In analogy with the Coulomb branch of the N = 4 super Yang-Mills theory, we argue that the discrete Higgs vacua of the mABJM theory as well as the corresponding LLM geometries are parametrized by the vevs of the chiral primary operators.

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