Quantum black holes, localization and the topological string

TL;DR

This work presents a concrete, exact macroscopic computation of quantum black hole entropy in ${\cal N}=2$ supergravity by applying supersymmetric localization to the AdS$_{2}$ near-horizon background. The authors show that the string-field functional integral localizes onto a finite-dimensional submanifold parameterized by real moduli ${C^{I}}$, yielding a simple renormalized action connected to the classical entropy function, and a final integral over ${C^{I}}$ with known corrections from brane instantons and one-loop determinants. They demonstrate how the leading piece aligns with the topological string via ${|Z_{top}|^{2}}$, while also outlining nonperturbative orbifold saddles and their impact on the full degeneracy. The framework provides a principled route to incorporate perturbative and nonperturbative corrections, and to relate macroscopic degeneracies to topological-string data and DT invariants. Overall, the paper establishes a robust connection between quantum entropy, localization, and topological strings in the ${\rm AdS}_{2}/{\rm CFT}_{1}$ context, with potential checks against known microscopic results in ${\cal N}=4$ cases.

Abstract

We use localization to evaluate the functional integral of string field theory on AdS_2xS^2 background corresponding to the near horizon geometry of supersymmetric black holes in 4d compactifications with N=2 supersymmetry. In particular, for a theory containing n_v + 1 vector multiplets, we show that the functional integral localizes exactly onto an ordinary integral over a finite-dimensional submanifold in the field space labeling a continuous family of instanton solutions in which auxiliary fields in the vector multiplets are excited with nontrivial dependence on AdS_2 coordinates. These localizing solutions are universal in that they follow from the off-shell supersymmetry transformations and do not depend on the choice of the action. They are parametrized by n_v +1 real parameters C^I, ; I= 0,..., n_v that correspond to the values of the auxiliary fields at the center of AdS_2. In the Type-IIA frame, assuming D-terms evaluate to zero on the solutions for reasons of supersymmetry, the classical part of the integrand equals the absolute square of the partition function of the topological string as conjectured by Ooguri, Strominger, and Vafa; however evaluated at the off-shell values of scalar fields at the center of AdS_2. In addition, there are contributions from one-loop determinants, brane-instantons, and nonperturbative orbifolds that are in principle computable. These results thus provide a concrete method to compute exact quantum entropy of these black holes including all perturbative and nonperturbative corrections and can be used to establish a precise relation between the quantum degeneracies of black holes and the topological string.