Large Order Enumerative Geometry, Black Holes and Black Rings
Abstract
Exploiting newly available data on Gopakumar-Vafa invariants at high genus for one-parameter hypergeometric Calabi-Yau threefolds, we study numerically the growth of the 5D indices, stable pair (PT) invariants and rank one Donaldson-Thomas (DT) invariants at large charges. For the 5D index Ω5D(d,m), below a critical value of the angular momentum m, we find perfect agreement with the Bekenstein-Hawking-Wald entropy of rotating 5D BMPV black holes, including the subleading correction from four-derivative interactions. When m exceeds the critical value, the 5D index is instead dominated by black rings with the smallest possible dipole charge. The stable pair invariant PT(d,m), which is determined by 5D indices, has a similar black ring/hole transition at negative m (now interpreted as the D0-brane charge) but surprisingly exhibits two other phase transitions at positive m: first, to a plateau and then to a polynomial growth ∼ m2d−1. In each phase, we derive an approximate expression for the invariant. Finally, the rank one DT invariant DT(d,m) is similar to PT(d,m) at negative m, and then transitions to a phase dominated by D0-branes, with entropy of order m2/3. Along the way, we determine the fixed genus, large degree behavior of GV invariants (including the overall g-dependent constant), extend it to an approximate formula valid also for large g, point out the unreasonable effectiveness of a simple PT/MSW relation, and study the growth of topological free energies at fixed degree, confirming a conjecture of Mariño.
Related
- Enumerative geometry of conics and double covers on projective complete intersections
- Quantum index, Arnold-Rokhlin surfaces, and real enumerative geometry
- Kerr black holes without primary hairs
- Spacetime Bartnik Mass Positivity and Temporal Monotonicity for Black Holes
- Perturbations of Kerr-Newman black holes: separation and mode stability
BibTeX
@misc{alexandrov2026large,
title = {{Large Order Enumerative Geometry, Black Holes and Black Rings}},
author = {Sergei Alexandrov and Albrecht Klemm and Boris Pioline},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.19552},
eprint = {2605.19552},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.19552},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Large Order Enumerative Geometry, Black Holes and Black Rings
AU - Sergei Alexandrov
AU - Albrecht Klemm
AU - Boris Pioline
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2605.19552
UR - https://arxiv.org/abs/2605.19552
AB - Exploiting newly available data on Gopakumar-Vafa invariants at high genus for one-parameter hypergeometric Calabi-Yau threefolds, we study numerically the growth of the 5D indices, stable pair (PT) invariants and rank one Donaldson-Thomas (DT) invariants at large charges. For the 5D index Ω_5D(d,m), below a critical value of the angular momentum m, we find perfect agreement with the Bekenstein-Hawking-Wald entropy of rotating 5D BMPV black holes, including the subleading correction from four-derivative interactions. When m exceeds the critical value, the 5D index is instead dominated by black rings with the smallest possible dipole charge. The stable pair invariant PT(d,m), which is determined by 5D indices, has a similar black ring/hole transition at negative m (now interpreted as the D0-brane charge) but surprisingly exhibits two other phase transitions at positive m: first, to a plateau and then to a polynomial growth ∼ m^2d−1. In each phase, we derive an approximate expression for the invariant. Finally, the rank one DT invariant DT(d,m) is similar to PT(d,m) at negative m, and then transitions to a phase dominated by D0-branes, with entropy of order m^2/3. Along the way, we determine the fixed genus, large degree behavior of GV invariants (including the overall g-dependent constant), extend it to an approximate formula valid also for large g, point out the unreasonable effectiveness of a simple PT/MSW relation, and study the growth of topological free energies at fixed degree, confirming a conjecture of Mariño.
ER -