Exact Holography of the Mass-deformed M2-brane Theory

Abstract

We test the holographic relation between the vacuum expectation values of gauge invariant operators in $\mathcal{N}=6$ ${\mathrm{U}}_k(N)\times {\mathrm{U}}_{-k}(N)$ mass-deformed ABJM theory and the LLM geometries with ${\mathbb{Z}}_k$ orbifold in 11-dimensional supergravity. To do that, 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 $\Delta =1$, which is given by $⟨{\mathcal{O}}^{(\Delta =1)}⟩=N^{\frac{3}{2}}{f}_{(\Delta =1)}$, for large $N$ and $k=1$. Here factor $f_{(\Delta )}$ is independent of $N$. Our results involve infinite number of exact dual relations for all possible supersymmetric Higgs vacua and so provide a nontrivial test of gauge/gravity duality away from the conformal fixed point. We also extend our results to the case of $k\ne 1$ for LLM geometries represented by rectangular-shaped Young-diagrams.

6 pages, major corrections in section 3 and 4, references added, title changed