# Exact holography of the mass-deformed M2-brane theory

The European Physical Journal C, May 2017

We test the holographic relation between the vacuum expectation values of gauge invariant operators in ${\mathcal {N}} = 6$ U$_k(N)\times \mathrm{U}_{-k}(N)$ mass-deformed ABJM theory and the LLM geometries with ${\mathbb {Z}}_k$ orbifold in 11-dimensional supergravity. To do so, 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 $\langle {\mathcal {O}}^{(\Delta =1)}\rangle = N^{\frac{3}{2}} \, f_{(\Delta =1)}$, for large N and $k=1$. Here the factor $f_{(\Delta )}$ is independent of N. Our results involve an infinite number of exact dual relations for all possible supersymmetric Higgs vacua and so provide a non-trivial test of gauge/gravity duality away from the conformal fixed point. We extend our results to the case of $k\ne 1$ for LLM geometries represented by rectangular-shaped Young diagrams. We also discuss the exact mapping of the gauge/gravity at finite N for classical supersymmetric vacuum solutions in field theory side and corresponding classical solutions in gravity side.

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Dongmin Jang, Yoonbai Kim, O-Kab Kwon, D. D. Tolla. Exact holography of the mass-deformed M2-brane theory, The European Physical Journal C, 2017, 342, DOI: 10.1140/epjc/s10052-017-4909-3