Exact holography of the mass-deformed M2-brane theory

Abstract

Abstract We test the holographic relation between the vacuum expectation values of gauge invariant operators in $${\mathcal {N}} = 6$$ N = 6 U $$_k(N)\times \mathrm{U}_{-k}(N)$$ k ( N ) × U - k ( N ) mass-deformed ABJM theory and the LLM geometries with $${\mathbb {Z}}_k$$ 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 $$\Delta = 1$$ Δ = 1 , which is given by $$\langle {\mathcal {O}}^{(\Delta =1)}\rangle = N^{\frac{3}{2}} \, f_{(\Delta =1)}$$ ⟨ O ( Δ = 1 ) ⟩ = N 3 2 f ( Δ = 1 ) , for large N and $$k=1$$ k = 1 . Here the factor $$f_{(\Delta )}$$ f ( Δ ) 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 $$k\ne 1$$ k ≠ 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.