Black String Entropy and Fourier-Mukai Transform

TL;DR

This work addresses microstate counting for black strings with D6–D2–D0 charge on elliptic Calabi–Yau threefolds by constructing a duality web that includes a relative Fourier–Mukai transform along elliptic fibers. This transform maps D6–D2–D0 configurations to D4–D2–D0 configurations, enabling a Cardy-based microscopic entropy calculation that is then matched to a macroscopic entropy computed from six-dimensional black-string solutions in N=1 supergravity. The leading entropy terms agree in the large-charge limit, with subleading corrections discussed in terms of curvature couplings and a proposed charge shift, highlighting the role of six-dimensional physics in correctly capturing the microstate count. Overall, the paper validates the Fourier–Mukai transform as a robust, autoequivalence-based tool for black hole microstate counting in F-theory contexts and clarifies the appropriate macroscopic framework (6D black strings) for D6-bearing configurations.

Abstract

We propose a microscopic description of black strings in F-theory based on string duality and Fourier-Mukai transform. These strings admit several different microscopic descriptions involving D-brane as well as M2 or M5-brane configurations on elliptically fibered Calabi-Yau threefolds. In particular our results can also be interpreted as an asymptotic microstate count for D6-D2-D0 configurations in the limit of large D2-charge on the elliptic fiber. The leading behavior of the microstate degeneracy in this limit is shown to agree with the macroscopic entropy formula derived from the black string supergravity solution.