Mass-deformed ABJM Theory and LLM Geometries: Exact Holography

Abstract

We present a detailed account and extension of our claim in arXiv:1610.01490. We test the gauge/gravity duality between the $\mathcal{N}=6$ mass-deformed ABJM theory with U${}_k(N)\times$U${}_{-k}(N)$ gauge symmetry and the 11-dimensional supergravity on LLM geometries with SO(4)/${\mathbb{Z}}_k$ $\times$SO(4)/${\mathbb{Z}}_k$ 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 $\Delta =1$. We show that $⟨{\mathcal{O}}^{(\Delta =1)}⟩=N^{\frac{3}{2}}{f}_{(\Delta =1)}$ for all supersymmetric vacuum solutions and LLM geometries with $k=1$, where the factor $f_{(\Delta )}$ is independent of $N$. We also confirm that the vacuum expectation value of the the energy momentum tensor is vanishing as expected by the supersymmetry. We extend our results to the case of $k\ne 1$ for LLM geometries represented by rectangular-shaped Young-diagrams. In analogy with the Coulomb branch of the $\mathcal{N}=4$ 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.

arXiv comment44 pages, 1 figure, major corrections in section 3 and 5, references added, title changed

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• Renormalized Holographic Subregion Complexity under Relevant Perturbations
• Holography of Massive M2-brane Theory with Discrete Torsion
• Undecidability in Spacetime Geometry via the AdS/CFT Correspondence