Euclidean analysis of the entropy functional formalism

TL;DR

The paper establishes a precise link between Sen's entropy functional formalism and the Euclidean zero-temperature approach to extremal black holes, showing that the on-shell entropy function f equals the zero-temperature Euclidean action and that near-horizon fields correspond to the dual CFT chemical potentials. By splitting the Euclidean action into near-horizon and asymptotic parts, it demonstrates that extremal solutions without ergoregions are fully captured by near-horizon data, reproducing Sen's results, while extremal non-BPS geometries with ergoregions exhibit subtle asymptotic contributions. A detailed dictionary between the two formalisms is developed, with explicit mapping in the D1-D5-P system and extensions to other black hole solutions. The work thus provides a unified framework linking attractor mechanism, Euclidean thermodynamics, and holographic interpretations of black hole entropy, and it discusses implications for higher-derivative corrections and CFT duals.

Abstract

The attractor mechanism implies that the supersymmetric black hole near horizon solution is defined only in terms of the conserved charges and is therefore independent of asymptotic moduli. Starting only with the near horizon geometry, Sen's entropy functional formalism computes the entropy of an extreme black hole by means of a Legendre transformation where the electric fields are defined as conjugated variables to the electric charges. However, traditional Euclidean methods require the knowledge of the full geometry to compute the black hole thermodynamic quantities. We establish the connection between the entropy functional formalism and the standard Euclidean formalism taken at zero temperature. We find that Sen's entropy function 'f' (on-shell) matches the zero temperature limit of the Euclidean action. Moreover, Sen's near horizon angular and electric fields agree with the chemical potentials that are defined from the zero-temperature limit of the Euclidean formalism.