Localization & Exact Holography

TL;DR

This paper uses localization techniques to exactly compute the bulk partition function in AdS_2/CFT_1 holography for supersymmetric black holes, matching boundary degeneracies and enabling quantum entropy calculations.

Contribution

It applies localization to reduce the bulk path integral to finite integrals, achieving an exact match with boundary degeneracies and advancing the proof of 'exact holography' for black holes.

Findings

Bulk partition function matches boundary degeneracies.

Exact evaluation reproduces Rademacher expansion terms.

Enables computation of quantum corrections to black hole entropy.

Abstract

We consider the AdS_2/CFT_1 holographic correspondence near the horizon of big four-dimensional black holes preserving four supersymmetries in toroidally compactified Type-II string theory. The boundary partition function of CFT_1 is given by the known quantum degeneracies of these black holes. The bulk partition function is given by a functional integral over string fields in AdS_2. Using recent results on localization we reduce the infinite-dimensional functional integral to a finite number of ordinary integrals over a space of localizing instantons. Under reasonable assumptions about the relevant terms in the effective action, these integrals can be evaluated exactly to obtain a bulk partition function. It precisely reproduces all terms in the exact Rademacher expansion of the boundary partition function as nontrivial functions of charges except for the Kloosterman sum which can in principle follow from an analysis of phases in the background of orbifolded instantons. Our results can be regarded as a step towards proving `exact holography' in that the bulk and boundary partition functions computed independently agree for finite charges. Since the bulk partition function defines the quantum entropy of the black hole, our results enable the evaluation of perturbative as well as nonperturbative quantum corrections to the Bekenstein-Hawking-Wald entropy of these black holes.