A Farey tale for N=4 dyons

TL;DR

The paper analyzes exponentially suppressed corrections to extremal black hole degeneracies within Sen's quantum entropy function, focusing on configurations with an intermediate $AdS_3$ region. It identifies an infinite family of $AdS_2$ saddles arising as extremal limits of the $AdS_3$ Farey tail, labeled by coprime pairs $(c,d)$, contributing terms $\exp(\mathcal{S}_0/c)$ with a phase $\exp(2\pi i q d/c)$, and shows these saddles lift to extremal BTZ geometries. In ${\cal N}=4$ theories, it then derives a Farey tale expansion for the dyon partition function by analyzing the pole structure of the Siegel modular form $\mathcal{Z}=1/\Phi_{10}$ and expressing the $4$D degeneracies as a Poincaré-type sum over poles in the Siegel upper half-plane, connecting to Jacobi and Hilbert modular forms. The results provide a macroscopic interpretation of subleading corrections as sums over $AdS_2$ geometries and establish a formal lift from Hilbert to Siegel modular forms with a pole on the diagonal divisor, linking the microscopic dyon degeneracies to the elliptic genus of the dual SCFT and the Dedekind/Igusa form structure. Together, these insights deepen the understanding of black hole entropy beyond the leading saddle and illuminate the intricate number-theoretic structure underlying ${\cal N}=4$ dyons.

Abstract

We study exponentially suppressed contributions to the degeneracies of extremal black holes. Within Sen's quantum entropy function framework and focusing on extremal black holes with an intermediate AdS3 region, we identify an infinite family of semi-classical AdS2 geometries which can contribute effects of order exp(S_0/c), where S_0 is the Bekenstein-Hawking-Wald entropy and c is an integer greater than one. These solutions lift to the extremal limit of the SL(2,Z) family of BTZ black holes familiar from the "black hole Farey tail". We test this understanding in N=4 string vacua, where exact dyon degeneracies are known to be given by Fourier coefficients of Siegel modular forms. We relate the sum over poles in the Siegel upper half plane to the Farey tail expansion, and derive a "Farey tale" expansion for the dyon partition function. Mathematically, this provides a (formal) lift from Hilbert modular forms to Siegel modular forms with a pole at the diagonal divisor.