Scaling Black Holes and Modularity

TL;DR

This paper develops techniques to compute partition functions for scaling black holes with multiple constituents in supergravity, revealing their mock modular form structure and invariance under dualities, with explicit examples in string theory.

Contribution

It introduces a method to determine partition functions of scaling black holes with magnetic charges, showing they form mock modular forms of depth two and analyzing their duality invariances.

Findings

Partition functions are mock modular forms of depth two.

S-duality invariance of the spectrum is demonstrated.

Explicit evaluation of partition functions in K3 fibrations.

Abstract

Scaling black holes are solutions of supergravity with multiple black hole singularities, which can be adiabatically connected to a single center black hole solution. We develop techniques to determine partition functions for such scaling black holes, if each constituent carries a non-vanishing magnetic charge corresponding to a D4-brane in string theory, or equivalently M5-brane in M-theory. For three constituents, we demonstrate that the partition function is a mock modular form of depth two, and we determine the appropriate non-holomorphic completion using generalized error functions. From the four-dimensional perspective, the modular parameter is the axion-dilaton, and our results show that $S$-duality leaves this subset of the spectrum invariant. From the five-dimensional perspective, the modular parameter is the complex structure of a torus $T^2$, and the scaling black holes are dual to states in the dimensional reduction of the M5-brane worldvolume theory to $T^2$. As a case study, we specialize the compactification manifold to a K3 fibration, and explicitly evaluate holomorphic parts of partition functions.