Holography of Massive M2-brane Theory with Discrete Torsion

Abstract

We investigate the gauge/gravity duality between the $\mathcal{N}=6$ 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)/${\mathbb{Z}}_k\times$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.

25 pages, 2 figures, The subleading contributions from the field theory calculations are reinterpreted