Black hole partition functions and duality

TL;DR

The paper develops a variational framework for black hole entropy in four-dimensional $N=2$ supergravity with higher-derivative and non-holomorphic corrections, showing the macroscopic entropy is the Legendre transform of a real free energy and can be recast via a real Hesse potential. It introduces duality-invariant partition functions and a hierarchy of inverse Laplace representations for microscopic degeneracies, and demonstrates consistency with known CHL degeneracies in the large-horizon regime, including a crucial measure factor that enforces duality covariance. The construction clarifies how non-holomorphic corrections and dualities (S- and T-duality) shape OSV-type relations and the relation to topological strings, while identifying persistent challenges for small, classically vanishing-area black holes. Overall, the work strengthens the microscopic/macroscopic bridge for BPS black holes in string theory, emphasizing the role of duality-covariant measures in black hole partition functions and delineating the limits of semiclassical agreement.

Abstract

The macroscopic entropy and the attractor equations for BPS black holes in four-dimensional N=2 supergravity theories follow from a variational principle for a certain `entropy function'. We present this function in the presence of R^2-interactions and non-holomorphic corrections. The variational principle identifies the entropy as a Legendre transform and this motivates the definition of various partition functions corresponding to different ensembles and a hierarchy of corresponding duality invariant inverse Laplace integral representations for the microscopic degeneracies. Whenever the microscopic degeneracies are known the partition functions can be evaluated directly. This is the case for N=4 heterotic CHL black holes, where we demonstrate that the partition functions are consistent with the results obtained on the macroscopic side for black holes that have a non-vanishing classical area. In this way we confirm the presence of a measure in the duality invariant inverse Laplace integrals. Most, but not all, of these results are obtained in the context of semiclassical approximations. For black holes whose area vanishes classically, there remain discrepancies at the semiclassical level and beyond, the nature of which is not fully understood at present.