# Holography of Massive M2-brane Theory: Non-linear Extension

## TL;DR

This paper extends the holographic correspondence between mass-deformed ABJM theory and 11D supergravity by implementing non-linear KK mappings, enabling the calculation of vacuum expectation values of operators and confirming supersymmetry predictions.

## Contribution

It provides a non-linear extension of KK holography for LLM geometries, allowing for more accurate gauge/gravity duality analysis in the mass-deformed ABJM context.

## Key findings

- Calculated VEVs of dimension 2 operators in large N limit.
- Confirmed vanishing VEVs of massive KK graviton modes due to supersymmetry.
- Demonstrated the importance of non-linear KK maps in holographic correspondence.

## Abstract

We investigate the gauge/gravity duality between the ${\cal N} = 6$ mass-deformed ABJM theory with U$_k(N)\times$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, in terms of a KK holography, which involves quadratic order field redefinitions. As a non-trivial extension of the previous work arXiv:1612.05066, we implement the KK mappings for various gauge invariant fields up to quadratic order to obtain 4-dimensional gravity fields. The non-linearity of the KK maps indicates that, in the current case, we can observe the true purpose of the non-linear KK holography of the LLM solutions. Using such KK holography procedure, we obtain the vacuum expectation values of the chiral primary operator of conformal dimension $\Delta = 2$ in the large $N$ limit but with general $k$ and examine the gauge/gravity duality for LLM solutions, which are represented by square-shaped Young diagrams. We also show that the vacuum expectation values of the massive KK graviton modes are vanishing as expected by the supersymmetry.