Holography of Massive M2-brane Theory with Discrete Torsion

We investigate the gauge/gravity duality between the ${cal N} = 6$ mass-deformed ABJ theory with U$_k(N+l)times$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 and the discrete torsion $l$. For chiral primary operators with conformal dimensions $Delta=1,2$, we obtain the exact vacuum expectation values using the holographic method in 11-dimensional supergravity and show that the results depend on the shapes of droplet pictures in LLM geometries. The $frac{l}{sqrt{N}}$ contributions from the discrete torsion $l$ for several simple droplet pictures in the large $N$ limit are determined in holographic vacuum expectation values. We also explore the effects of the orbifolding ${mathbb Z}_k$ and the asymptotic discrete torsion $l$, on the gauge/gravity duality dictionary and on the nature of the asymptotic limits of the LLM geometries.