Localization in Supergravity and Quantum $AdS_4/CFT_3$ Holography

TL;DR

The paper develops a bulk localization computation for quantum gravity in $AdS_4/CFT_3$ by using a four-dimensional gauged supergravity truncation with a square-root prepotential, reducing the M-theory partition function on $AdS_4\times X_7$ to a finite-dimensional integral over two localizing instanton moduli. The renormalized off-shell action is linear and holomorphic in the prepotential at the AdS center, and after holographic renormalization and a flux term, the bulk integral reproduces the Airy function that matches the ABJM $S^3$ partition function at large $N$, up to assumptions about the measure and ignoring brane-instanton corrections. The work connects bulk localization to the boundary Airy structure and hints at deeper links between the prepotential in gauged supergravity and topological-string-like data, suggesting future exploration of nonperturbative completions and refined topological-string interpretations. Overall, the approach provides a nonperturbatively exact bulk computation in four-dimensional Planck units within a controlled truncation and clarifies the holographic origin of the Airy function in this $AdS_4/CFT_3$ setting.

Abstract

We compute the quantum gravity partition function of M-theory on $AdS_4 \times X_7 $ by using localization techniques in four-dimensional gauged supergravity obtained by a consistent truncation on the Sasaki-Einstein manifold $X_{7}$. The supergravity path integral reduces to a finite dimensional integral over two collective coordinates that parametrize the localizing instanton solutions. The renormalized action of the off-shell instanton solutions depends linearly and holomorphically on the "square root" prepotential evaluated at the center of $AdS_{4}$. The partition function resembles the Laplace transform of the wave function of a topological string and with an assumption about the measure for the localization integral yields an Airy function in precise agreement with the computation from the boundary ABJM theory on a 3-sphere. Our bulk quantum gravity computation is nonperturbatively exact in four-dimensional Planck length but ignores corrections due to brane-instantons.