Higher Derivative Corrections to Non-supersymmetric Extremal Black Holes in N=2 Supergravity

TL;DR

This work applies the entropy function formalism to compute the entropy of extremal black holes in four-dimensional ${ m N}=2$ supergravity with higher-derivative (curvature-squared) corrections, covering both supersymmetric and non-supersymmetric configurations. It develops the formalism in general, demonstrates symplectic invariance, and provides explicit results in the STU model and in M-theory compactifications on Calabi-Yau manifolds, including leading and first-order corrections due to higher-derivative terms. A central finding is a discrepancy between the four-dimensional results and five-dimensional expectations for a class of non-supersymmetric black holes, suggesting that the minimal four-dimensional supersymmetric completion misses terms present after dimensional reduction from five dimensions. The results highlight limitations of the four-dimensional curvature-squared action and motivate the search for a complete higher-derivative action that faithfully captures higher-dimensional origin and duality relations, especially for non-supersymmetric cases.

Abstract

Using the entropy function formalism we compute the entropy of extremal supersymmetric and non-supersymmetric black holes in N=2 supergravity theories in four dimensions with higher derivative corrections. For supersymmetric black holes our results agree with all previous analysis. However in some examples where the four dimensional theory is expected to arise from the dimensional reduction of a five dimensional theory, there is an apparent disagreement between our results for non-supersymmetric black holes and those obtained by using the five dimensional description. This indicates that for these theories supersymmetrization of the curvature squared term in four dimension does not produce all the terms which would come from the dimensional reduction of a five dimensional action with curvature squared terms.