Macroscopic entropy formulae and non-holomorphic corrections for supersymmetric black holes

TL;DR

This work develops a framework to compute and reconcile macroscopic black-hole entropy in four-dimensional $N=2$ (and $N=4$) compactifications with subleading microscopic corrections. It demonstrates how higher-derivative $C^2$ terms and non-holomorphic corrections to the effective couplings are essential for duality-invariant entropy formulas, and shows how to extend macroscopic results to dual Type-II/M-theory descriptions. A key finding is that consistent microstate counting for $K3\times T^2$ requires a modification of zero-mode counting, aligning the microscopic entropy with the corrected macroscopic entropy. The paper also explores non-perturbative and GV-type corrections to the entropy, showing how non-holomorphic completions and logarithmic terms restore duality invariance across the full moduli space.

Abstract

In four-dimensional N=2 compactifications of string theory or M-theory, modifications of the Bekenstein-Hawking area law for black hole entropy in the presence of higher-derivative interactions are crucial for finding agreement between the macroscopic entropy obtained from supergravity and subleading corrections to the microscopic entropy obtained via state counting. Here we compute the modifications to the area law for various classes of black holes, such as heterotic black holes, stemming from certain higher-derivative gravitational Wilsonian coupling functions. We consider the extension to heterotic N=4 supersymmetric black holes and their type-II duals and we discuss its implications for the corresponding micro-state counting. In the effective field theory approach the Wilsonian coupling functions are known to receive non-holomorphic corrections. We discuss how to incorporate such corrections into macroscopic entropy formulae so as to render them invariant under duality transformations, and we give a concrete example thereof.