Exact Holography of the Mass-deformed M2-brane Theory

TL;DR

This paper establishes an exact holographic relation between vacuum expectation values of certain operators in mass-deformed ABJM theory and LLM geometries, providing a precise test of gauge/gravity duality beyond the conformal point.

Contribution

It derives an exact holographic relation for VEVs in mass-deformed ABJM theory and extends the analysis to nontrivial LLM geometries with orbifold and Higgs vacua.

Findings

Exact relation for =1 operator VEVs at large N and k=1

Validation of gauge/gravity duality away from conformal fixed point

Extension of results to k geometries with rectangular Young diagrams

Abstract

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.