Supersymmetry, Localization and Quantum Entropy Function

TL;DR

The work addresses computing black hole state degeneracies via the quantum entropy function in the $AdS_2/CFT_1$ framework. It uses enhanced supersymmetry and localization to show that only $H_1$-invariant string configurations contribute to the near-horizon path integral, while infinite zero modes from AdS$_2$ asymptotics yield finite results. The analysis identifies $H_1$-invariant saddles, including freely acting ${\mathbb{Z}}_s$ orbifolds, with classical actions scaling as $\exp(S_{\rm Wald}/s)$ and demonstrates finite contributions from the superconformal current algebra zero modes. This provides a controlled macroscopic computation of entropy that can be matched with microscopic degeneracies and offers a route to finite-dimensional reductions and extensions to related observables in holography.

Abstract

AdS_2/CFT_1 correspondence leads to a prescription for computing the degeneracy of black hole states in terms of path integral over string fields living on the near horizon geometry of the black hole. In this paper we make use of the enhanced supersymmetries of the near horizon geometry and localization techniques to argue that the path integral receives contribution only from a special class of string field configurations which are invariant under a subgroup of the supersymmetry transformations. We identify saddle points which are invariant under this subgroup. We also use our analysis to show that the integration over infinite number of zero modes generated by the asymptotic symmetries of AdS_2 generate a finite contribution to the path integral.