Partition function of non-supersymmetric black holes in the supergravity limit

TL;DR

This work addresses the problem of formulating a mixed partition function for non-supersymmetric black holes arising from CY3 compactifications in type II string theory within the supergravity limit. It reformulates the attractor equations into a C-based framework and extends the OSV mixed-ensemble construction to the non-supersymmetric sector via a reduced ensemble and Legendre transforms, yielding an explicit ns-BH partition function. The main result is an explicit expression for the non-susy partition function, Z_ns.BH(D(p), phi0) = exp(pi^2 D(p)/(2 phi0)), along with its charge-space interpretation and a discussion of symplectic considerations and limitations. This provides a structured path toward counting non-BPS microstates in the supergravity regime, while highlighting open issues such as extensions beyond the supergravity limit and the role of D-terms and holomorphic anomalies.

Abstract

In this note, we propose the free energy of general \emph{non-supersymmetric} black hole attractors arising in type IIA(B) superstrings on 3-fold Calabi-Yau, in the supergravity limit. This, by definition, differs from its counterpart BPS free energy by a factor of $4 $. Correspondingly, a mixed ensemble for these black holes is proposed.