Euclidean analysis of the entropy functional formalism

The attractor mechanism implies that the supersymmetric black hole near horizon solution is defined in terms of the conserved charges only and therefore is 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 so far little understood 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.