Scaling Black Holes and Modularity
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 nonvanishing magnetic charge corresponding to a D4brane in string theory, or equivalently M5brane in Mtheory. For three constituents, we demonstrate that the partition function is a mock modular form of depth two, and we determine the appropriate nonholomorphic completion using generalized error functions. From the fourdimensional perspective, the modular parameter is the axiondilaton, and our results show that $S$duality leaves this subset of the spectrum invariant. From the fivedimensional 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 M5brane 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.05504
 Bibcode:
 2021arXiv211005504C
 Keywords:

 High Energy Physics  Theory;
 Mathematics  Number Theory
 EPrint:
 52 pages