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A Farey Tail for Attractor Black Holes

Jan de Boer, Miranda C. N. Cheng, Robbert Dijkgraaf, Jan Manschot, Erik Verlinde

TL;DR

This work extends the Black Hole Farey Tail to four-dimensional attractor black holes by formulating the BPS degeneracies as a generalized elliptic genus of a (0,4) CFT arising from wrapped M5-branes on a Calabi–Yau. Using spectral flow and a vector-valued Rademacher (Farey tail) expansion, the authors decompose the elliptic genus into holomorphic characters and theta functions, obtaining an exact semi-classical expansion in terms of AdS$_3$ saddles and a polar part controlled by Gopakumar–Vafa invariants. The gravitational interpretation interprets the modular sum as a sum over Euclidean AdS$_3$ fillings (BTZ/thermal AdS) with subleading contributions from a gas of wrapped M2-branes, linking to the OSV conjecture and topological string data via $|Z_ ext{top}|^2$. The results provide a precise framework connecting microstate counting to macroscopic geometries and topological invariants, while highlighting truncations and avenues for refinement through a full Poincaré-series treatment. Key takeaways include a concrete attractor Farey Tail formula and its physical interpretation in terms of M-theory brane dynamics and topological strings.

Abstract

The microstates of 4d BPS black holes in IIA string theory compactified on a Calabi-Yau manifold are counted by a (generalized) elliptic genus of a (0,4) conformal field theory. By exploiting a spectral flow that relates states with different charges, and using the Rademacher formula, we find that the elliptic genus has an exact asymptotic expansion in terms of semi-classical saddle-points of the dual supergravity theory. This generalizes the known "Black Hole Farey Tail" of [1] to the case of attractor black holes.

A Farey Tail for Attractor Black Holes

TL;DR

This work extends the Black Hole Farey Tail to four-dimensional attractor black holes by formulating the BPS degeneracies as a generalized elliptic genus of a (0,4) CFT arising from wrapped M5-branes on a Calabi–Yau. Using spectral flow and a vector-valued Rademacher (Farey tail) expansion, the authors decompose the elliptic genus into holomorphic characters and theta functions, obtaining an exact semi-classical expansion in terms of AdS saddles and a polar part controlled by Gopakumar–Vafa invariants. The gravitational interpretation interprets the modular sum as a sum over Euclidean AdS fillings (BTZ/thermal AdS) with subleading contributions from a gas of wrapped M2-branes, linking to the OSV conjecture and topological string data via . The results provide a precise framework connecting microstate counting to macroscopic geometries and topological invariants, while highlighting truncations and avenues for refinement through a full Poincaré-series treatment. Key takeaways include a concrete attractor Farey Tail formula and its physical interpretation in terms of M-theory brane dynamics and topological strings.

Abstract

The microstates of 4d BPS black holes in IIA string theory compactified on a Calabi-Yau manifold are counted by a (generalized) elliptic genus of a (0,4) conformal field theory. By exploiting a spectral flow that relates states with different charges, and using the Rademacher formula, we find that the elliptic genus has an exact asymptotic expansion in terms of semi-classical saddle-points of the dual supergravity theory. This generalizes the known "Black Hole Farey Tail" of [1] to the case of attractor black holes.

Paper Structure

This paper contains 15 sections, 105 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Wrapped M-branes and the Near Horizon Limit
  3. Wrapped branes on Calabi-Yau and the spectral flow
  4. The near-horizon geometry and reduction to three dimensions
  5. The (0,4) SCFT and its Elliptic Genus
  6. The (0,4) superconformal field theory
  7. A generalized elliptic genus
  8. The Farey Tail Expansion of the Elliptic Genus
  9. The Rademacher formula
  10. Application to the (0,4) elliptic genus
  11. Spacetime Interpretation of the Attractor Farey Tail
  12. Gravitational interpretation of the Rademacher formula
  13. Interpretation in terms of M2-branes
  14. Discussion
  15. Restoring the $\overline{\tau}$-dependence

Figures (3)

  • Figure 1: An M5 brane loops through the original (circular) M5 brane and then annihilates again with an anti-M5 brane.
  • Figure 2: A large gauge transformation: (i) An M5-anti-M5 pair wrapping the equator of the $S^2$ nucleates at the center of the $AdS_2$. (ii) The M5 and anti-M5 begin to move in the opposite directions in the $S^2$, while still stay at the center of the $AdS_2$. (iii) A Dirac surface wrapping the whole $S^2$ is formed. (iv) Finally one moves the Dirac surface from the bulk of the $AdS_2$ towards the spatial infinity across the boundary.
  • Figure 3: (i) Thermal $AdS_3$, with the B-cycle being contractible. (ii) BTZ black hole, with the A-cycle being contractible. (iii) The geometry with the (A+B)-cycle being contractible.