F-Theory, Spinning Black Holes and Multi-string Branches

TL;DR

The paper develops a detailed framework linking 6d strings from F-theory to 5d spinning BPS black holes, using D3-branes on genus-g curves and their elliptic-genus 2d CFTs with an SU(2)_L current algebra. It uncoveres a multi-branch structure of IR CFTs arising from curve degenerations, with bound states formed via KK momentum and intricate pole structures in meromorphic Jacobi forms that govern degeneracies. The authors connect microscopic counts to macroscopic entropy through topological strings and AdS_3×S^3 holography, deriving central charges and levels that match subleading gravitational corrections and expose a spectrum including J^2 > M^3 configurations bound by angular momentum in Taub-NUT setups. They provide concrete analyses for M-theory on elliptic P^2 with explicit d-level curves, illustrating how multi-string branches and pole interpretations shape the black hole entropy across phases. The work highlights a rich, modular- and Jacobi-form–driven landscape of bound states beyond the single-centered black hole, with potential implications for 4d black holes and M5-brane bound states in broader Calabi-Yau contexts.

Abstract

We study 5d supersymmetric black holes which descend from strings of generic $\mathcal{N}=(1,0)$ supergravity in 6d. These strings have an F-theory realization in 6d as D3 branes wrapping smooth genus $g$ curves in the base of elliptic 3-folds. They enjoy $(0,4)$ worldsheet supersymmetry with an extra $SU(2)_L$ current algebra at level $g$ realized on the left-movers. When the smooth curves degenerate they lead to multi-string branches and we find that the microscopic worldsheet theory flows in the IR to disconnected 2d CFTs having different central charges. The single string sector is the one with maximal central charge, which when wrapped on a circle, leads to a 5d spinning BPS black hole whose horizon volume agrees with the leading entropy prediction from the Cardy formula. However, we find new phenomena where this branch meets other branches of the CFT. These include multi-string configurations which have no bound states in 6 dimensions but are bound through KK momenta when wrapping a circle, as well as loci where the curves degenerate to spheres. These loci lead to black hole configurations which can have total angular momentum relative to a Taub-Nut center satisfying $J^2 > M^3$ and whose number of states, though exponentially large, grows much slower than those of the large spinning black hole.

Figures (9)

• Figure 1: The entropy of a spinning black hole corresponding to large charge $Q$ with $SU(2)_L$ spin $J$ is schematically plotted here for fixed charge $Q$ as a function of $J$. We see that the states $J^2 \gg Q^3$ have an entropy which grows linearly with $\sqrt{Q^2}$ independent of $J$. These large values of $J$ are due to large orbital angular momentum. For black holes arising from strings in 6d, $\sqrt{Q^3-J^2}\sim \sqrt{C^2n-J^2}$ where $C$ denotes the string charge and $n$ is the momentum around the circle and the large angular momentum states scale as $\sqrt{Q^2}\sim \sqrt{Cn}$.
• Figure 2: The single string branch is connected to multi-string branches in the UV. In the IR they lead to disconnected CFT's. However, there are additional contributions to the elliptic genus of the full CFT where the branches meet which correspond to multi-string states which are bound by KK-momenta. The effective central charge of these bound strings is expected to be the same as that of the multi-string branches.
• Figure 3: D3-branes wrapping curves in K3 give rise to strings in 6d. Upon wrapping a circle in a compactification to five dimensions several strings can join to form a multi-wound string.
• Figure 4: The toric skeleton of a degree $d=3$ hypersurface in $\mathbb{P}^2$. The number of lines ending on each edge indicates the degree of the curve in such a description. One can see that $d\geq 3$ curves necessarily form a genus $g \geq 1$ Riemann surface (in this case $g=1$ as one can see from the single hole in the middle of the graph).
• Figure 5: The toric skeleton of a degree $d=2$ hypersurface in $\mathbb{P}^2$ undergoing a phase transition. The left-most figure shows the generic case of a degree 2 curve. The central figure is a degeneration to 2 degree 1 curves, each wrapped by a D3 brane giving rise to strings which are depicted as points in $\mathbb{R}^4$. The last picture shows the Higgs branch where the two strings are separated in $\mathbb{R}^4$.