Subleading and non-holomorphic corrections to N=2 BPS black hole entropy

TL;DR

This paper analyzes subleading and non-holomorphic corrections to the entropy of four-dimensional N=2 BPS black holes by introducing a real, homogeneous Omega that encodes non-holomorphic contributions to the effective action. It shows that duality invariance requires a nontrivial integration measure in the mixed electric/magnetic partition function and derives a semiclassical, OSV-like expression with determinants that include the Omega-dependent corrections. The authors compute explicit forms for Omega^(1) in the FHSV and STU models, compare the resulting macroscopic entropies with topological-string expectations, and discuss the extent to which Omega corrections can be interpreted as non-holomorphic deformations of special geometry. In the STU model they further examine small and large black holes, compare to the proposed microstate degeneracies, and identify subtleties in reconciling microscopic formulas with macroscopic attractor dynamics, underscoring a nuanced relationship between non-holomorphic corrections, duality, and topological-string data.

Abstract

BPS black hole degeneracies can be expressed in terms of an inverse Laplace transform of a partition function based on a mixed electric/magnetic ensemble, which involves a non-trivial integration measure. This measure has been evaluated for black holes with various degrees of supersymmetry and for N=4 supersymmetric black holes all results agree. It generally receives contributions from non-holomorphic corrections. An explicit evaluation of these corrections in the context of the effective action of the FHSV model reveals that these are related to, but quantitatively different from, the non-holomorphic corrections to the topological string, indicating that the relation between the twisted partition functions of the latter and the effective action is more subtle than has so far been envisaged. The effective action result leads to a duality invariant BPS free energy and arguments are presented for the existence of consistent non-holomorphic deformations of special geometry that can account for these effects. A prediction is given for the measure based on semiclassical arguments for a class of N=2 black holes. Furthermore an attempt is made to confront some of the results of this paper to a recent proposal for the microstate degeneracies of the STU model.