Black Hole Entropy, Special Geometry and Strings

TL;DR

This review analyzes macroscopic and microscopic entropy of four-dimensional ${ m N}=2$ black holes arising from string and M-theory compactifications, emphasizing higher-curvature corrections and the Wald entropy framework. It develops a comprehensive off-shell superconformal approach to ${ m N}=2$ supergravity, linking special Kähler geometry, symplectic dualities, and the prepotential $F(X^I,A)$ to the near-horizon attractor mechanism. The authors derive a model-independent entropy formula that remains covariant under dualities and shows how the stabilization equations fix horizon moduli in terms of charges, enabling quantitative matches with microscopic state counts in Calabi–Yau compactifications. The work further discusses Type II, M-theory, and heterotic realizations, the role of $R^2$-terms, and the duality-invariant structure of the entropy, highlighting the deep connections between black hole thermodynamics, topological string theory, and the geometry of the compactified dimensions.

Abstract

We review work done over the last years on the macroscopic and microscopic entropy of supersymmetric black holes in fourdimensional N=2 supergravity and in N=2 compactifications of string theory and M-theory. Particular emphasis is put on the crucial role of higher curvature terms and of modifications of the area law in obtaining agreement between the macroscopic entropy, which is a geometric property of black hole solutions and the microscopic entropy, which is computed by state counting in Calabi-Yau compactifications of string or M-theory. We also discuss invariance properties of the entropy under stringy T-duality and S-duality transformations in N=2,4 compactifications in presence of higher curvature terms. In order to make the paper self-contained we review the laws of black hole mechanics in higher derivative gravity, the definition of entropy as a surface charge, the superconformal off-shell description of N=2 supergravity, special geometry, and N=2 compactifications of heterotic and type II string theory and of M-theory.