Exact holography of the mass-deformed M2-brane theory

Abstract

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 kne 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.