We test the holographic relation between the vacuum expectation values of gauge invariant operators in ${ \cal N} = 6$ ${ \rm U}_k(N)\times { \rm U}_{ -k}(N)$ 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 $\langle { \cal O}^{ (\Delta=1)}\rangle= N^{ \frac32} \, f_{ (\Delta=1)}$, 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.
Categories: hep-th