Classical and quantum N=2 supersymmetric black holes

TL;DR

The paper addresses how entropy formulas for N=2, D=4 black holes in heterotic and type-II string vacua can incorporate quantum corrections. It develops the N=2 prepotential framework to study central-charge extremization and derives a perturbative heterotic entropy formula in which the tree-level form is preserved with the perturbative coupling ${g^2_{ m pert}}$ replacing the bare coupling, yielding ${ m S}/ ext{π}=rac{8 extπ}{g_{ m pert}^2}ig|_{ m hor}raket{N,N}$. For type-II compactifications on Calabi–Yau three-folds, new entropy/area relations are obtained that depend on the electric/magnetic charges and topological CY data ${C_{ABC}}$, ${c_2 ext{·}J_A}$, and the Euler characteristic $ extχ$, including axion-free reductions and a symplectic shift that accommodates a linear prepotential term. The work also bridges these 4D results to higher-dimensional geometries via M-theory, interpreting the black holes as configurations of intersecting M5-branes and showing how horizon data emerge from CY topology. Overall, the results provide a more complete understanding of how quantum corrections and topology shape black-hole entropy in N=2 theories, with potential implications for non-perturbative quantum gravity in string theory.

Abstract

We use heterotic/type-II prepotentials to study quantum/classical black holes with half the $N=2, D=4$ supersymmetries unbroken. We show that, in the case of heterotic string compactifications, the perturbatively corrected entropy formula is given by the tree-level entropy formula with the tree-level coupling constant replaced by the perturbative coupling constant. In the case of type-II compactifications, we display a new entropy/area formula associated with axion-free black-hole solutions, which depends on the electric and magnetic charges as well as on certain topological data of Calabi--Yau three-folds, namely the intersection numbers, the second Chern class and the Euler number of the three-fold. We show that, for both heterotic and type-II theories, there is the possibility to relax the usual requirement of the non-vanishing of some of the charges and still have a finite entropy.