Black Holes and Large Order Quantum Geometry

TL;DR

The paper tests a GV/DT-based microscopic description of five-dimensional black holes arising from M-theory on Calabi–Yau threefolds, against their macroscopic entropy, using advanced topological-string methods on one-parameter models and K3-fibered geometries. It demonstrates that GV invariants account for the macroscopic entropy of large 5d black holes, reveals universal cancellations consistent with OSV-type relations, and provides exact degeneracy formulas and asymptotics for small black holes via heterotic/type II duality. The work combines high-genus topological-string data with Richardson transforms to extract large-charge asymptotics, analyzes DT invariants to establish a universal growth exponent k ≈ 2, and delivers closed-form asymptotics for microstate counts in K3 fibrations, clarifying the interplay between microscopic counting, modular structure, and macroscopic gravity. Overall, it strengthens the link between enumerative geometry invariants and black-hole thermodynamics in string theory.

Abstract

We study five-dimensional black holes obtained by compactifying M theory on Calabi-Yau threefolds. Recent progress in solving topological string theory on compact, one-parameter models allows us to test numerically various conjectures about these black holes. We give convincing evidence that a microscopic description based on Gopakumar-Vafa invariants accounts correctly for their macroscopic entropy, and we check that highly nontrivial cancellations -which seem necessary to resolve the so-called entropy enigma in the OSV conjecture- do in fact occur. We also study analytically small 5d black holes obtained by wrapping M2 branes in the fiber of K3 fibrations. By using heterotic/type II duality we obtain exact formulae for the microscopic degeneracies in various geometries, and we compute their asymptotic expansion for large charges.

Figures (12)

• Figure 1: Microscopic data for $f(d)$ ($\Box$), and the Richardson transforms $A(d,2)$ ($\triangle$), $A(d,3)$ ($\diamond$), and $A(d,4)$ ($\star$). The straight line corresponds to the macroscopic prediction $b_0=\frac{4\pi}{3\sqrt{2\kappa}}$. For the quintic this value is $b_0\approx 1.359$ and for the available degree $14$ the Richardson transforms lie $1.8$, $2.1$, $1.2$ % from the macroscopic prediction. For the bi-cubic $b_0\approx 0.967$, the available degree is higher, $18$, and the microscopic counting is within $.9$, $1.2$, $.3$ % from the macroscopic prediction. As an example we give BPS numbers used for the analysis at degree $18$ of the bi-cubic in table (A.1).
• Figure 2: Microscopic data for $f(d)$ ($\Box$), and the Richardson transforms $A(d,4)$ ($\triangle$), $A(d,5)$ ($\diamond$), and $A(d,6)$ ($\star$). The straight line corresponds to the macroscopic prediction $b_1=\frac{\pi c_2}{4\sqrt{2\kappa}}$. For the degree $X_{6,2}$ complete intesection this value is $b_1\approx 14.44$ and for the available degree $12$ the Richardson transforms lie $-11.7$, $-10.4$, $-9.77$ % below the macroscopic prediction. For the bi-cubic $b_1\approx 9.994$, the available degree is $18$ and the microscopic counting is $-7.15$, $-6.88$, $-6.63$ % below the macroscopic prediction.
• Figure 3: The plot of $-b_2$ vs. $(-\chi\kappa^{\frac{1}{6}})$ for 13 Calabi-Yau models.
• Figure 4: Castelnuovo's bound for higher genus curves on the quintic. The dots represent $n_{d}^{g_{\rm max}}$ and the curve is (\ref{['bound']}).
• Figure 5: Scaling data $k^{(0)}$ ($\Box$) and the transforms $k^{(1)}$ ($\triangle$), $k^{(2)}$ ($\diamond$) for the Donaldson-Thomas invariants on the quintic in $\mathbb{P}^4$ starting for $(d,0)$ states.