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We compute the quantum gravity partition function of M-theory on $AdS_4\times X_7 $ by using localization techniques in four-dimensional gaugedsupergravity obtained by a consistent truncation on the Sasaki-Einsteinmanifold $X_{7}$. The supergravity path integral reduces to a finitedimensional integral over two collective coordinates that parametrize thelocalizing instanton solutions. The renormalized action of the off-shellinstanton solutions depends linearly and holomorphically on the "square root"prepotential evaluated at the center of $AdS_{4}$. The partition functionresembles the Laplace transform of the wave function of a topological stringand with an assumption about the measure for the localization integral yieldsan Airy function in precise agreement with the computation from the boundaryABJM theory on a 3-sphere. Our bulk quantum gravity computation isnonperturbatively exact in four-dimensional Planck length but ignorescorrections due to brane-instantons.

Localization in Supergravity and Quantum $AdS_4/CFT_3$ Holography