# Holography of Massive M2-brane Theory with Discrete Torsion

## TL;DR

This paper explores the holographic duality between a mass-deformed ABJ theory with discrete torsion and 11D supergravity on LLM geometries, deriving exact vacuum expectation values and analyzing the effects of orbifolding and torsion.

## Contribution

It provides exact holographic vacuum expectation values for chiral primary operators and examines how discrete torsion and orbifolding influence the gauge/gravity duality in LLM geometries.

## Key findings

- Exact VEVs depend on droplet shapes in LLM geometries.
- Discrete torsion contributes terms proportional to l/√N in large N limit.
- Orbifolding and torsion affect the duality dictionary and asymptotic geometry limits.

## Abstract

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)/${\mathbb Z}_k\times$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.