Properties of Black Holes in Toroidally Compactified String Theory

TL;DR

This work surveys the macroscopic and microscopic properties of black holes in toroidally compactified string theories across dimensions $4\le D\le 9$, focusing on generating solutions created by dualities and their BPS limits. It derives entropy formulas for $D=4$, $D=5$, and $5<D<10$, discusses duality-invariant representations, and connects these classical results to microscopic counts from elementary strings, quantum hair, and D-branes, including the Strominger–Vafa paradigm. The authors then present a unifying M-theory perspective by interpreting these black holes as intersecting M-branes, showing how harmonic and non-extremality functions encode their structure. The work highlights that the macroscopic entropy exhibits forms that align with microscopic degeneracies in near-BPS and non-extremal regimes, supporting a deep link between gravity and string/M-theory microphysics with potential broader applicability beyond strictly BPS configurations.

Abstract

We review the macroscopic and microscopic properties of black holes of toroidally compactified heterotic and Type II string theory in dimensions 4<=D<=9. General charged rotating black hole solutions are obtained by acting on a generating solution with classical duality symmetries. In D=4, D=5 and 6<=D<=9, the generating solution for both toroidally compactified Type II and heterotic strings is specified by the ADM mass, [(D-1)/2]-angular momentum components and five, three and two charges, respectively. We give the Bekenstein-Hawking entropy for these solutions, address the BPS-saturated limit and compare the results to calculations of the microscopic entropy both in the NS-NS sector and the R-R sector of the theory. We also interpret such black hole solutions as dimensionally reduced intersecting p-branes of M-theory.